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Perimeter: Problems with Solutions

Perimeter is the distance around a two-dimensional shape.

Example: the perimeter of this rectangle is 7+3+7+3 = 20

Example: the perimeter of this regular pentagon is:.

3 + 3 + 3 + 3 + 3 = 5×3 = 15

Perimeter Formulas

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Perimeter of Rectangle Problems

Perimeter of a rectangle word problems.

There are different types of geometry word problems. One of the most common involves the perimeter which is the distance around the outside of a two-dimensional geometric shape. In this lesson, our focus is on the perimeter of a rectangle in which we’ll find the relationship between the sides of a rectangle, that is, the length and the width.

The key to solving perimeter word problems is to know two things:

1) The formula of the perimeter of a rectangle.

• Perimeter of a Rectangle Formula: [latex]{\textbf{\textit{P = 2L + 2W}}}[/latex]
• Remember that the length (L) is the longest side of the rectangle while the width (W) is the shortest side .

2) Being able to express the length in terms of the width and vice versa, depending on the word problem.

Let’s tackle this topic by working on a variety of examples involving the perimeter of a rectangle.

Just a heads up, the first five examples (word problems 1-5) can be solved by multi-step equations with one or single variable while the last three (word problems 6-8), can be solved by systems of equations with two variables.

Example 1: The perimeter of a rectangle is 128 centimeters and the length is 47 centimeters. Find the width of the rectangle.

The first thing that we need to do is to construct or draw a diagram based on the information given in the problem. Looking back at our problem, we were given the measurement for the length as well as the perimeter of the rectangle.

With this visual, we can easily spot what pieces of information are already provided to us and what else is missing that we need to solve. Clearly, the width is unknown so we’ll represent this value with the variable [latex]W[/latex].

Let’s now solve for the width ([latex]W[/latex]) using the formula for the perimeter of a rectangle. Since we already know the value for the length ([latex]L[/latex]) and the value for the perimeter ([latex]P[/latex]), we will simply replace the variables ([latex]L[/latex]) and ([latex]P[/latex]) with their values.

Answer: The width of the rectangle is 17 centimeters.

Before we proceed to the next problem, let’s find out if our answer is correct. We can do this by simply substituting the values that we have for the perimeter, length, and width into the perimeter formula, then verify if each side of the equation equals the other.

Great! We’re able to confirm that the measurement of the width which is 17 cm, indeed is the correct value.

Example 2: A rectangle has a width of 7 feet and a length of 132 inches. Find the perimeter of the rectangle both in feet and inches.

This problem is asking us to express the perimeter of the rectangle using two different units of measurement, i.e. in feet and in inches. To get both measurement units, we’ll solve the problem in two parts.

PART 1: Express the perimeter of the rectangle in feet.

We are given the following information below:

Since the width is already in feet, we don’t have to do anything with it. However, for the length, we must convert 132 inches to feet. We know that 1 ft = 12 in.

Perfect! We were able to get the measurement of our length in feet. So now, we have:

• Width = 7 ft
• Length = 11 ft

To find the perimeter, we simply have to plug in the values into the Perimeter Formula then simplify.

Part 1 Answer: The perimeter of the rectangle is 36 feet.

PART 2: Express the perimeter of the rectangle in inches.

We will use the same methodology in Part 1 to find our answer for this second part. Again, the following pieces of information are given to us in the original problem:

• Width = 7 ft.
• Length = 132 in.

This time, the length is the side that has its measurement already in inches. Since we want to find the rectangle’s perimeter in inches, we’ll focus on converting the width from feet to inches. Remember again that 12 in. = 1 ft .

So now we have,

The dimensions of our length and width are now both in inches. Let’s now find the perimeter of the rectangle by substituting the values into the perimeter formula then simplify.

Part 2 Answer: The perimeter of the rectangle is 432 inches.

Example 3: The width of a rectangle is 23 meters less than the length. The perimeter of the rectangle is 94 meters. Find the dimensions of the rectangle.

For us to solve this word problem, we need to be able to substitute values or expressions to the [latex]L[/latex], [latex]W[/latex], and [latex]P[/latex] variables of the perimeter formula, [latex]P=2L+2W[/latex].

In our problem, we can easily spot the value of the perimeter ([latex]P[/latex]) which is 94 meters. On the other hand, the width of the rectangle is expressed in terms of length.

The width of the rectangle is 23 meters less than the length.

We need to translate the algebraic sentence above into an algebraic expression so we can substitute it for the width in the perimeter formula. So if ([latex]W[/latex]) stands for the width and ([latex]L[/latex]) is for the length, our algebraic expression for the width is,

[latex]{W = L – 23}[/latex]

How about the length? Well, at this time, our length is unknown. So in this case, we will keep the variable [latex]L[/latex] to stand for the length.

Let’s construct a diagram to get a better visual of all the information that we have so far.

Now that it’s clear to us what we’ll substitute the [latex]L[/latex], [latex]W[/latex], and [latex]P[/latex] variables with, we can proceed to solve for [latex]L[/latex] using the perimeter of a rectangle formula.

Since the problem is asking us to find the dimensions for both the length and the width, let’s use the value of [latex]L[/latex] to also get the value for the width.

• Length ([latex]L[/latex])= [latex]35[/latex]
• Width ([latex]W[/latex])= [latex]L-23={\color{red}35}-23=12[/latex]

Answer: The length of the rectangle is 35 meters and the width is 12 meters.

I advise my students to always check their answers no matter how confident they feel that they got the correct one. For this problem, we simply have to substitute the values of [latex]L[/latex], [latex]W[/latex], and [latex]P[/latex] in the perimeter formula. If the left and the right side of the equation both equal each other, then our answers are correct.

Example 4: The length of the rectangle is 12.5 feet more than the width. If the perimeter of the rectangle is 101 feet, how long is the length of the rectangle?

Right off the bat, we can see that the value of the perimeter is 101 ft and that the length is defined using the width as given in the following statement.

The length of the rectangle is 12.5 feet more than the width.

Just like in our previous example, we need to algebraically express the length in terms of the width so we can substitute this algebraic expression for the length ([latex]L[/latex]) in the perimeter formula. Translating the statement above, we get [latex]L = W+12.5[/latex]

Let’s put together the information that we’ve gathered so far in a diagram.

As you can see, the width ([latex]W[/latex]) is currently unknown. Therefore, the next step that we have to do is solve for ([latex]W[/latex]) using the formula for the perimeter of a rectangle.

Now that we know the value of the width ([latex]W[/latex]), let’s use this value to find the dimension for the length.

• Width ([latex]W[/latex])= [latex]19[/latex]
• Length ([latex]L[/latex])= [latex]W+12.5={\color{red}19}+12.5=31.5[/latex]

Going back to our original problem, how long is the length of the rectangle?

Answer: The length of the rectangle is 31.5 ft.

This time, I will leave it up to you to check if our answer is correct. Start by substituting the values of [latex]L[/latex], [latex]W[/latex], and [latex]P[/latex] in the perimeter formula, [latex]P=2L+2W[/latex], then simplify. If both sides of the equation equal each other, then we got the correct answer.

Example 5: A certain rectangle has a length of half a yard more than twice the width. The perimeter of the unknown rectangle is 55 yards. Find its width.

Let’s start by determining the values and expressions that are provided to us in the problem.

• Length = half a yard more than twice the width = [latex]2W +\Large {1 \over 2}[/latex]
• Perimeter = 55

Using the perimeter formula, we will solve for the unknown which is the width ([latex]W[/latex]).

Before we formally answer the question from our word problem, let’s find out the dimensions of the rectangle using the value of the width.

• Width ([latex]W[/latex])= [latex]9[/latex]
• Length ([latex]L[/latex])= [latex]2W +{\Large {1 \over 2}}=2({\color{red}9})+{\Large {1 \over 2}}=18+{\Large {1 \over 2}}=18{\Large{1 \over 2}}[/latex]

Answer: The width of the rectangle is 9 yards.

Example 6: When you subtract the width from the length of the rectangle, the difference is 8 inches. What are the dimensions of the rectangle if its perimeter is 72 inches?

This example is a little bit more complex than the previous ones. As you may have noticed, neither the length nor the width is expressed in terms of the other. So to proceed, we will start by discussing the important statements that are given to us in this problem.

When you subtract the width from the length of the rectangle, the difference is 8 inches .

Let’s use again the variable [latex]L[/latex] to stand for length and [latex]W[/latex] for the width. Translating this algebraically, we have the following equation

[latex]L – W = 8[/latex]

The perimeter of the rectangle is 72 inches.

We are already familiar with the formula for the perimeter of a rectangle so we can simply write this as

[latex]P = 2L + 2W[/latex]

[latex]72 = 2L + 2W[/latex]

By just going through both statements, we were able to come up with two equations:

• Equation 1: [latex]L – W = 8[/latex]
• Equation 2: [latex]72 = 2L + 2W[/latex]

It is clear at this point that we are dealing with two equations with two unknowns, namely, [latex]L[/latex] and [latex]W[/latex]. In other words, we are going to solve systems of linear equations with two variables.

Observe that we can use Equation 1 to express the length in terms of the width, then substitute the expression into Equation 2. The result will be a multi-step equation having the width ([latex]W[/latex]) as the only variable in the equation.

Let’s do this step-by-step.

1) Start with Equation 1 and solve for [latex]L[/latex].

Notice that to clean up the left side of the equation, we added [latex]W[/latex] to both sides of the equation. Then we applied the Commutative Property of Addition, that is, [latex]8 + W = W + 8[/latex].

2) Next, substitute the expression of [latex]L[/latex] into Equation 2.

We now have the value for the width. But since we are asked to find both dimensions of the rectangle, let’s plug the value of the width which is 14 into Equation 1 to find the value for the length ([latex]L[/latex]).

So, what are the dimensions of the rectangle if its perimeter is 72 inches?

Answer: The length of the rectangle is 22 inches while the width is 14 inches.

Calculator Check:

Example 7: The difference of the length and three times the width of a rectangle is 5 centimeters. Find the length and the width of the rectangle if its perimeter is 82 centimeters.

We have the same situation here as our previous example. Neither the length nor the width is expressed in terms of the other. Therefore, let’s break down the problem again to find out what important pieces of information are given to us.

First, we are told that

The difference between the length and three times the width of a rectangle is 5 centimeters.

Using [latex]L[/latex] and [latex]W[/latex] again as our unknown variables, we can translate this algebraic sentence as

[latex]L – 3W = 5[/latex]

Next, our second statement says that

The perimeter of the rectangle is 82 centimeters.

This one is easy. Using the formula for the perimeter of a rectangle we have,

[latex]P = 2L + 2W\,\,\, \to \,\,\,82 = 2L + 2W[/latex]

Before we proceed, let’s look at our two equations:

• Equation 1: [latex]L – 3W = 5[/latex]
• Equation 2: [latex]82 = 2L + 2W[/latex]

Our next step is to express [latex]L[/latex] in terms of the width.

1) Solve for [latex]L[/latex] using Equation 1.

We are now ready to plug in the expression of [latex]L[/latex] into our second equation.

2) Substitute [latex]L[/latex] with [latex]3W + 5[/latex] in Equation 2.

Let’s now also see what the value of the length is using the value we got for the width which is 9.

Answer: The length of the rectangle is 32 centimeters and the width is 9 centimeters.

Let’s verify again real quick using a calculator if both values will give us a perimeter of 82 cm when plugged into the formula of the perimeter of a rectangle. If the sum of twice the length ([latex]2L[/latex]) and twice the width ([latex]2W[/latex]) is 82, then both of our answers are correct.

Example 8: The sum of the length and one-half of the width is 42 yards. The rectangle’s perimeter is 100 yards. What is the width of the rectangle?

You should be familiar by now with what to do first when we have perimeter word problems where neither each side of the rectangle is defined using the other.

Let’s delve into the important statements right away and translate them into an algebraic equation.

The sum of the length and one-half of the width is 42 yards.

• Equation 1: [latex]L + {\Large{1 \over 2}}W = 42[/latex]

The rectangle’s perimeter is 100 yards.

• Equation 2: [latex]P = 2L + 2W\,\,\, \to \,\,\,100 = 2L + 2W[/latex]

What’s next? Well, it’s time for us to define the length using the width by solving for [latex]L[/latex] using Equation 1.

Let’s now substitute [latex]L[/latex] with [latex]42 – {\Large{1 \over 2}}W[/latex] in Equation 2.

The word problem is only asking us for the measurement of the width. So we’ll say,

Answer: The width of the rectangle is 16 yards.

I’ll leave it up to you to do the checking if our answer is correct. Make sure to find the value of the length first by using the value of the width ([latex]W[/latex]) which is 16.

Then, move on to check if both values, when plugged into the perimeter of a rectangle formula ([latex]P=2L+2W[/latex]), will give you a perimeter of 100 yards as stated in our original word problem.

You might also like these tutorials:

• Perimeter of a Rectangle
• Area of a Rectangle

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How to find perimeter

Here you will learn how to find perimeter, including what the perimeter of a shape is, how to calculate it for different shapes, and how to solve perimeter word problems.

Students will first learn how to find perimeter as part of measurement and data in 3rd grade and 4th grade.

What is the perimeter?

The perimeter of a 2D shape is the total distance around the outside of the shape. 

For example,

Perimeter is measured in units. Each side of the square is one unit.

The perimeter of the rectangle is found by counting the units on each side of the rectangle. You can do this by starting at a vertex and counting each unit around the shape until you arrive back at the vertex.

There are 20 units around the outside of the rectangle, so the perimeter is 20 units.

Notice the perimeter can also be found by adding the length of each side.

6 + 4 + 6 + 4 = 20, so the perimeter is 20 units.

This addition strategy can be used to find the perimeter for any polygon.

Find the perimeter of the triangle.

\text{Perimeter }=5+12+13=30 \ \text{ cm}

Find the perimeter of the parallelogram.

\text{Perimeter }= 8 + 4 + 8 + 4 = 24 units

Note, whenever a specific unit is not given (like cm, m, or km ), the perimeter is labeled ‘units.’

Common Core State Standards

How does this relate to 3rd grade math and 4th grade math?

• Grade 3 – Measurement and data (3.MD.D.8) Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters.
• Grade 4 – Measurement and data (4.MD.A.3) Apply the area and perimeter formulas for rectangles in real world and mathematical problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor.

To find the perimeter of a shape:

Add all the side lengths.

Write the final answer with the correct units.

[FREE] Perimeter Check for Understanding Quiz (Grade 3 to 4)

Use this quiz to check your grade 3 to 4 students’ understanding of perimeter. 10+ questions with answers covering a range of 3rd and 4th grade perimeter topics to identify areas of strength and support!

How to find perimeter examples

Example 1: gridded rectangle.

What is the perimeter of the rectangle?

The length of the rectangle is 7 units.

The width of the rectangle is 5 units.

The opposite sides of the rectangle are congruent (equal).

To find the perimeter (the total distance around the rectangle), add all the side lengths:

7 + 5 + 7 + 5 = 24

2 Write the final answer with the correct units.

There are no specific units given, so they are just labeled ‘units.’

The perimeter of the rectangle is 24 units.

Example 2: given perimeter – find missing side length

The perimeter of the rectangle is 78 \, m. Find the missing side length.

The opposite sides of a rectangle are congruent (equal).

So, the perimeter is 34 \, + \, ? \, + \, 34 \, + \, ?=78 \, m.

Since 34 + 34 = 68, the two missing sides make up the rest of the perimeter.

Subtract to see how much more is needed to get to a total perimeter of 78 :

So, the combined length of the missing sides is 10.

You need to solve ? \, + \, ? = 10. Since the two sides are congruent, each side must be 5, because 5 + 5 = 10.

The rectangle is measured in meters (m).

The missing side length is 5 \, m.

Example 3: perimeter of isosceles triangle

What is the perimeter of the triangle?

To find the perimeter (the total distance around the triangle), add all the side lengths:

11 + 11 + 9 = 31

The triangle is measured in inches.

The perimeter of the triangle is 31 inches.

Example 4: perimeter of a trapezoid

What is the perimeter of the trapezoid?

To find the perimeter (the total distance around the trapezoid), add all the side lengths:

The trapezoid is measured in centimeters (cm).

The perimeter of the trapezoid is 24 \, cm.

Example 5: perimeter of rectilinear shape

What is the perimeter of the polygon?

When a shape is shown with a grid, the perimeter can be found by counting the total side lengths instead of adding them.

Choose a starting point and count each side around the shape until you reach the starting point again. Be sure to count each unit.

The model above labels each unit, but you don’t need to do this as you solve. The units can also be counted out loud.

Also, notice the star on the first unit. It can be helpful to mark where you start counting so you don’t forget where to stop.

The perimeter of the polygon is 36 units.

Example 6: perimeter of rectilinear shape

There are missing measurements in the polygon. You can use what you know about other sides to find the length of the sides.

Since 5 + 3 = 8, the missing measurement is 3 \, m.

To find the total length of the large rectangle, add 10 \, m and 13 \, m.

10 + 13 = 23

The missing length of the large rectangle is 23 \, m.

Now that all side lengths are known, add them all to find the perimeter.

8 + 10 + 3 + 13 + 5 + 23 = 62

The polygon is measured in meters (m).

The perimeter of the polygon is 62 \, m.

Teaching tips for how to find the perimeter

• Choose worksheets that provide a mixture of regular and irregular shapes, both gridded and non-gridded, so students can practice all perimeter solving strategies. This also challenges them to decide which strategies are useful for each type of shape.
• Encourage students to look for patterns or extend known patterns to include perimeter. For example, recognizing that all triangles will have 3 sides to add, and all quadrilaterals will have 4 sides to add, etc. This also includes patterns seen with the calculating perimeter formulas. For example, you may keep an anchor chart of equations used to find the area of different rectangles. If you consistently ask students to look for patterns, they will eventually notice that the same numbers are being added and some students will connect this to doubling or multiplying numbers. You can then use this as the basis for introducing the formula for the perimeter of a rectangle: 2l+2w =p.

Easy mistakes to make

• Thinking the order of adding the sides matters Since addition is commutative, it doesn’t matter the order in which the sides of the shape are added. Adding in any order will result in the same answer as long as all sides are added only once.
• Confusing the formulas or solving strategies for perimeter with area Since they are often introduced together, in the beginning it is easy to confuse the two vocabulary words and/or concepts. After enough exposure and intentional classroom discourse, students will understand and remember the difference.
• Confusing the units for perimeter with area If students do not have a complete understanding of why perimeter is units and area is square units, they may mix up when to use each. Providing opportunities such as physically measuring the perimeter of items and working with area on grids, can help build an understanding of the difference.
• Confusing the units In order to add the sides together, they need to be in the same units. If the sides are given in different units, convert to one common unit before adding.

Related perimeter lessons

• Perimeter of a square
• Perimeter of a rectangle
• Perimeter of a triangle

Practice how to find perimeter questions

1. What is the perimeter of the rectangle?

The perimeter is the total distance around the outside of the rectangle.

The length of the rectangle is 10 units.

The width of the rectangle is 2 units.

10 + 2 + 10 + 2 = 24

You can also count each unit of the rectangle to find the perimeter:

2. The perimeter of the rectangle is 94 \, cm. Find the missing side length.

So, the perimeter is ? \, + \, 21 \, + \, ? \, + \, 21 = 94 \, cm.

Since 21 + 21 = 42, the two missing sides make up the rest of the perimeter.

Subtract to see how much more is needed to get to a total perimeter of 94 :

So, the combined length of the missing sides is 52.

You need to solve ? \, + \, ? = 52.

Since the two sides are congruent, each side must be 26, because 26 + 26 = 52.

The missing side length is 26 \, cm.

3. What is the perimeter of the hexagon?

To find the perimeter (the total distance around the hexagon), add all the side lengths:

10 + 4 + 4 + 10 + 4 + 4 = 36

The hexagon is measured in feet (ft).

The perimeter of the hexagon is 36 \, ft.

4. What is the perimeter of the regular pentagon?

All sides of a regular pentagon are the same length.

To find the perimeter (the total distance around the pentagon), add all the side lengths:

7 + 7 + 7 + 7 + 7 = 35

The pentagon is measured in inches.

The perimeter of the pentagon is 35 inches.

5. What is the perimeter of the polygon?

The perimeter of the polygon is 28 units.

6. What is the perimeter of the polygon?

The missing side on the smaller rectangle is 18 \, ft.

To find the missing side, solve ? \, + \, 18 = 29.

The missing side must be 11, since 11 + 18 = 29.

To find the missing side, solve 6 \, + \, ? = 25.

The missing side must be 19, since 6 + 19 = 25.

Now that all side lengths are known, add them all to find the perimeter. Remember to only add the side lengths of the polygon.

6 + 29 + 25 + 11 + 19 + 18 = 108.

The polygon is measured in feet (ft).

The perimeter of the polygon is 108 \, ft.

How to find perimeter FAQs

In upper grades, students learn how to find the perimeter of a circle. They also learn how to use the distance formula to calculate the perimeter of regular and irregular polygons.

Perimeter is the distance around a rectangle and is measured in units. Area is the space within a rectangle and is measured in square units. They represent different parts of a rectangle, but both require knowing the dimensions of the length and width to solve.

Perimeter is the distance around a triangle and is measured in units (such as feet). Area is the space within a triangle and is measured in square units (such as square feet). For both, you need to know the base length, but other parts of the formulas vary.

The next lessons are

• Angles in polygons
• Congruence and similarity
• Prism shape

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Perimeter – Definition, Regular and Irregular Shapes, Examples

What is a perimeter, how to find perimeter, solved examples on perimeter, practice problems on perimeter, frequently asked questions on perimeter.

In geometry, the perimeter of a shape is defined as the total length of its boundary. The perimeter of a shape is determined by adding the length of all the sides and edges enclosing the shape. It is measured in linear units of measurement like centimeters, meters, inches, or feet.

Let’s try to calculate the perimeter of the following shape:

Perimeter of a shape = Sum of all its sides,

Perimeter of the given shape = 6 cm + 5 cm + 5 cm + 4 cm + 3 cm = 23 cm.

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Perimeter of a Regular Shape

We know that the length of each side of a regular polygon is the same. Therefore, Perimeter of regular polygon = sum of all its sides = number of sides ✕ length of one side.

For example, look at the given regular pentagon.

The perimeter of the given regular pentagon can be calculated as follows:

Number of sides = 5 Length of one side = 4 cm

Therefore, Perimeter = number of sides ✕ length of one side

= 5 ✕ 4 = 20 cm

The given table summarizes the formulas to find the perimeter of some regular polygons:

Perimeter of An Irregular Shape

Since the sides of an irregular polygon may not all be the same in length, we use the general formula to find the perimeter of an irregular shape. Therefore, Perimeter of irregular polygon = sum of all sides.

Real – World Applications

We often use the concept of perimeter in real life. For example, when putting up Christmas lights around the house or when we want to put a fence around the backyard, we find its perimeter to know the length of wire we will need.

Example 1: What is the perimeter of the given figure?

Solution:  

We know that the perimeter of a triangle is given by

Perimeter = a + b + c,

Where a, b, c = length of three sides.

For the given triangle,

Perimeter = 5 cm + 4 cm + 3 cm = 12 cm

Example 2: Calculate the perimeter of the following figure.

The given shape is an irregular pentagon.

The perimeter of this pentagon will be given by the sum of all its sides.

Perimeter = 2 cm + 3 cm + 3 cm + 4 cm + 5 cm = 17 cm

Example 3: What will be the perimeter of a rectangle with length 12 cm and breadth 5 cm?

We know that the perimeter of a rectangle is given by 

Perimeter = 2 ✕ (l + b),

Where l = length of the given rectangle, b = breadth of the given rectangle.

For the given rectangle, l = 12 cm, b = 5 cm.

Therefore, perimeter of given rectangle = 2 ✕ (12 + 5) = 2 ✕ 17 = 34 cm

Attend this Quiz & Test your knowledge.

Find the perimeter of a square that has sides of 40 cm in length each.

Calculate the perimeter of a rectangle with length = 30 cm and breadth = 14 cm., find the perimeter of a circle with radius = 7 cm., calculate the perimeter of the following figure:.

How many sides are required to determine the perimeter of a regular heptagon?

We only require the length of one side to determine the perimeter of any geometric figure with sides of equal length, such as a regular heptagon. The area of a heptagon = 7 ✕ a, where a is the side length.

What is the difference between the perimeter and area of a 2-D shape?

Perimeter measures the length of the boundary of the shape. It is given by the sum of all sides of the shape. It is one-dimensional and expressed in linear units. 

Area, on the other hand, measures the space occupied by the shape. It is two-dimensional and is expressed in square units.

How can we find the perimeter of a polygon?

The most general way to find the perimeter of any polygon is to find the sum of all its sides. However, another way of finding the perimeter of a regular polygon is to multiply the number of its sides by the side length.

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Perimeter Word Problems Worksheet

Related Topics & Worksheets: Math Worksheets Printable Math Worksheets Free Online Worksheet to help students practice solving perimeter word problems for regular shapes and for rectangles.

• I know how to solve perimeter word problems for regular shapes and for rectangles.

Perimeter refers to the total distance around the outside of a two-dimensional shape. It is calculated by adding up the lengths of all the sides of the shape.

A regular shape is a two-dimensional shape where all sides have equal length and all angles have equal measure. When calculating the perimeter of a regular shape we can either add the lengths of all the sides or we can find the length of one side and multiply by the number of sides.

When calculating the perimeter of a rectangle, we can add up the lengths of all the sides or we can use the formula P = 2(l + w) where P is the perimeter, l is the length, and w is the width. This formula works because a rectangle has two pairs of parallel sides, and each pair has the same length. Therefore, to find the perimeter, you can add up the lengths of all four sides, which is the same as adding up two lengths and two widths.

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Helping with Math

What is a perimeter?

Perimeter is a closed path that covers or surrounds any two-dimensional shape . We can say that a perimeter is the total distance of any closed shape. Here are the examples of shapes that we can determine its perimeter. 

How to find the perimeter?

Finding the perimeter of any two-dimensional shape depends on the number of sides our shape has. Say, for example, we have a triangle and a square. The number of sides a triangle has is 3; hence, we will get the total distance around its three sides. 

Now, how do we find the perimeter? Well, we need to start identifying the shape of our figure first, then use a formula in getting the perimeter. Each shape has a different formula in determining the perimeter – and that’s what we will tackle in the next sections.

The perimeter of a circle is called as the circumference . A circle is still considered a two-dimensional shape even though it does not have a length and width. However, they have radius and diameter . We can determine the circumference of any circle through its radius or diameter. 

Finding the Perimeter of a Circle through Radius

The radius is the distance from the center of the circle to any point on the circumference. The image shows the radius of a circle.  

Now, to get the circumference of a circle using its radius, we will use the formula

where C = circumference,

    = 3.14 or $\frac{22}{7}$

    r = radius

What is the circumference of a circle if the radius is 18 cm? 

Solution 

Find the total distance around the circle with a radius that measures 30 meters. 

Finding the Perimeter of a Circle through its Diameter

The diameter is the longest line segment in any circle. The measure of a diameter is twice the measure of a radius. Hence, the formula of finding the circumference of a circle through its diameter is given by:

= 3.14 or $\frac{22}{7}$

d = diameter

Example #1 

What is the circumference of a circle with a diameter of 13 mm? 

Determine the total distance around the circle when the diameter is 46 decimeters. 

A triangle is a two-dimensional closed shape with three straight sides. Every triangle has three sides, three vertices, and three angles . If given a triangle, we simply add all the measures of its sides. 

The formula in finding the perimeter of a triangle is given by 

P = a + b + c

where a, b, and c are the measures of the sides of a triangle. 

If the measures of the sides of a triangle are 17 cm, 23 cm, and 25 cm, what is its perimeter? 

Can you determine the perimeter of a triangle if all three sides measure 19 meters?

Quadrilaterals 

A quadrilateral is any polygon with exactly four sides. There are different kinds of quadrilaterals such as square, rhombus , rectangle, trapezoid , and rhombus. 

A square is a quadrilateral where each of its sides is equal. Hence, we can get the perimeter of a square by adding the measures of all four sides or by simply multiplying the measure of one side by 4.

Thus, the formula of finding the perimeter of the square is: 

where s is the measure of the side of a square. 

What is the perimeter of a square where each side measures 10 centimeters? 

Determine the total distance around a square where one side measures 27 meters.

Like squares, a rhombus is a quadrilateral where four sides are equal, and the pair of opposite sides are parallel to each other. Hence, the formula in finding the perimeter of a rhombus is the same as finding the perimeter of a square. 

Thus, it is given by the formula

where s is the measure of the side of a rhombus. 

What is the perimeter of a rhombus if the measure of one side is 9 units?

Find the total distance of a rhombus with a side measure of 35 kilometers.

A rectangle is a two-dimensional quadrilateral where opposite sides are parallel and equal. The sides of a rectangle are called as a width and length. Thus, the formula given in getting the perimeter of a rectangle is:

P = 2( l + w)

where l = length

What is the perimeter of a rectangle if the length measures 19 centimeters and the width measures 21 centimeters?

Determine the perimeter of a rectangle if the length measures 350 centimeters and the width measures 3 meters.

Regular Polygon

Regular polygons are two-dimensional geometric figures with a fixed number of sides and where all sides are equal. 

Hence, the formula of finding the perimeter of any regular polygon is given by:

where n = number of sides of a regular polygon

s = measure of the side of a regular polygon

What is the perimeter of a 9-sided polygon with side measures of 23 meters?

Determine the perimeter of a five-sided regular polygon where each side measures 11 centimeters. 

What is the formula for finding the perimeter of some closed-shape?

The table below shows the formula for finding the perimeter of some closed-shape.

How to solve problems involving perimeter?

To solve problems that involve finding the perimeter of any closed shape , you may follow the following steps:

• Identify the shape being described.
• Make sure that the unit of measurement is always the same.
• Use the formula for finding the perimeter of the given shape.
• Always remember to write the proper unit of measurement. 

A rectangular field garden measures 45 yards by 60 yards. What is the total distance that surrounds the garden?

The diameter of a clock measures 10 inches. What is the circumference of the clock?

A seven-sided regular polygon with a side measure of 13 decimeters is needed for Gwyneth’s project. What is the total distance that surrounds the polygon? 

What is the importance of finding the perimeter?

The use of perimeter is used in our daily lives, especially architects and engineers. Most people who understand the concept of the perimeter can help them design their room , remodel their kitchen , or even build a table or chair . Farmers, gardeners, or even lot owners also use their knowledge in perimeter to fence their lots. More so, perimeters are important in estimating and calculating materials needed for completing a certain projects.

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Perimeter Calculator

Table of contents

With this perimeter calculator, you don't need to worry about perimeter calculations anymore. Below you'll find the perimeter formulas for twelve different shapes, as well as a quick reminder about what a perimeter is and a perimeter definition. Read on, give it a try, or check this calculator's twin brother – our comprehensive area calculator .

What is perimeter?

Perimeter is the boundary of a closed geometric figure . It may also be defined as the outer edge of an area, simply the longest continuous line that surrounds a shape. The name itself comes from Greek perimetros : peri meaning "around" + metron , understood as "measure". As it's the length of the shape's outline, it's expressed in distance units – e.g., meters, feet, inches, or miles.

How to find perimeter – perimeter formulas

Usually, the most simple and straightforward approach is to find the sum of all of the sides of a shape . However, there are cases where there are no sides (such as an ellipse, circle, etc.), or one or more sides are unknown. In this paragraph, we'll list all of the equations used in this perimeter calculator.

Scroll down to the next sections if you're curious about a specific shape, and wish to see an explanation, derivation, and image for each of the twelve shapes present in this calculator. We also have tools dedicated to each shape – just type the name of the shape in the search bar at the top of this webpage.

Here are the perimeter formulas for the twelve geometric shapes in this calculator:

Square perimeter formula: P = 4 a P = 4a P = 4 a .

Rectangle perimeter formula: P = 2 ( a + b ) P = 2(a + b) P = 2 ( a + b ) .

Triangle perimeter formulas:

• P = a + b + c P = a + b + c P = a + b + c ; or
• P = a + b + a 2 + b 2 − 2 a b × cos ⁡ ( γ ) P = a + b + \sqrt{a^2 + b^2 - 2ab \times \cos(\gamma)} P = a + b + a 2 + b 2 − 2 ab × cos ( γ ) ​ ; or
• P = a + a sin ⁡ ( β + γ ) × ( sin ⁡ ( β ) + sin ⁡ ( γ ) ) P = a + \frac{a}{\sin(\beta + \gamma)} \times (\sin(\beta) + \sin(\gamma)) P = a + s i n ( β + γ ) a ​ × ( sin ( β ) + sin ( γ )) .

Circle perimeter formula: P = 2 π r P = 2\pi r P = 2 π r .

Circle sector perimeter formula: P = r ( α + 2 ) P = r(\alpha + 2) P = r ( α + 2 ) ( α \alpha α is in radians);

Ellipse perimeter formula: P = π [ 3 ( a + b ) − ( 3 a + b ) × ( a + 3 b ) ] P = \pi\bigl[3(a + b) - \sqrt{(3a + b) \times (a + 3b)}\bigl] P = π [ 3 ( a + b ) − ( 3 a + b ) × ( a + 3 b ) ​ ] ;

Quadrilateral / Trapezoid perimeter formula: P = a + b + c + d P = a + b + c + d P = a + b + c + d .

Parallelogram perimeter formulas:

• P = 2 ( a + b ) P = 2(a + b) P = 2 ( a + b ) ;
• P = 2 a + 2 e 2 + 2 f 2 − 4 a 2 P = 2a + \sqrt{2e^2 + 2f^2 - 4a^2} P = 2 a + 2 e 2 + 2 f 2 − 4 a 2 ​ ; or
• P = 2 ( b + h / sin ⁡ ( α ) ) P = 2(b + h/\sin(\alpha)) P = 2 ( b + h / sin ( α )) .

Rhombus perimeter formulas:

• P = 4 a P = 4a P = 4 a ; or
• P = 2 e 2 + f 2 P = 2\sqrt{e^2 + f^2} P = 2 e 2 + f 2 ​ .

Kite perimeter formula: P = 2 ( a + b ) P = 2(a + b) P = 2 ( a + b ) .

Annulus perimeter formula: P = 2 π ( R + r ) P = 2\pi(R + r) P = 2 π ( R + r ) .

Regular polygon perimeter formula: P = n × a P = n \times a P = n × a .

Perimeter of a square formula

A square has four sides of equal length. To calculate its perimeter, all you need to do is to multiply the side length by 4 4 4 :

Believe it or not, but we have a perimeter of a square calculator , too!

Formula for the perimeter of a rectangle

The formula for the perimeter of a rectangle is almost as easy as the equation for the perimeter of a square. The only difference is that we have two pairs of equal-length sides:

Perimeter of a triangle formula

The easiest formula to calculate the perimeter of a triangle is – as usual – by summing all sides:

However, you aren't always given three sides. What can you do then? Well, instead of fretting, you can use the law of cosines calculator to find the missing side:

This can be incorporated into the perimeter formula:

The other option is to use the law of sines if you have one side and the two angles that are adjacent to that side:

so the triangle perimeter may be expressed as:

Perimeter of a circle formula (circumference formula)

A perimeter of a circle has a special name – it's also known as the circumference . The most well-known perimeter of a circle formula uses only one variable – circle radius:

Have you ever wondered how many times your bike wheel will rotate on a ten-mile trip? Well, that's one of the cases where you'll need to use the circumference formula. Input the radius of your wheel (half of the wheel's diameter), and divide 10 miles by the obtained circumference (but don't forget about the conversion of the units of length!). If you want to be even more accurate, you can include the size of the bike tire.

Perimeter of a circle sector formula

Calculating the perimeter of a circle sector may sound tricky – is it only the arc length, or is it the arc length plus two radii? Just keep in mind the perimeter definition! The sector perimeter is the sum of the lengths of all its boundaries, so it's the latter:

where α \alpha α is in radians.

Perimeter of an ellipse formula (ellipse circumference formula)

Although the formula for the area of an ellipsis really simple and easy to remember, the perimeter of an ellipse formula is the most troublesome of all the equations listed here. We've chosen to implement one of the Ramanujan approximations in this perimeter calculator:

Where a a a is the shortest possible radius and b b b in the longest possible radius of an ellipse. The other, more accurate Ramanujan approximation is:

There is also a simpler form, using an additional variable h h h :

Or you could just use our calculator!

Perimeter of a trapezoid formula

If you want to calculate the perimeter of an irregular trapezoid, there's no special formula – just add all four sides:

Maybe you've noticed, but it's the formula for any quadrilateral perimeter.

There's also an option that presents itself with certain special trapezoids – like an isosceles trapezoid, where you need a a a , b b b , and c c c sides. Another example is a right trapezoid, where the length of the bases and one leg are enough to find the shape's perimeter (to find the last leg, we calculate Pythagoras' Theorem).

Perimeter of a parallelogram formula

In this perimeter calculator, you'll find three formulas to calculate the perimeter of a parallelogram:

• The most straightforward one, adding all sides together:
• The perimeter of a parallelogram formula that requires one side and diagonals
• The perimeter is given in terms of base, height, and any parallelogram angle .

Perimeter of a rhombus formula

The perimeter of a rhombus formula is not rocket science, so let's make it concise – it's the same as the perimeter of a square formula!

Another solution to finding the rhombus perimeter requires the diagonal lengths:

Try deriving the formula yourself. You know that the two diagonals of a rhombus are perpendicular to and bisect each other so that you can divide the shape into four congruent right triangles. Each triangle has legs that are e/2 and f/2 long – all you need to do is find the triangle's hypotenuse, which is, at the same time, the rhombus side. Then multiply the result by four to find the final perimeter of a rhombus formula.

Perimeter of a kite formula

The formula for the perimeter of a kite is pretty straightforward – just sum up all of the sides:

Perimeter of an annulus formula

As the perimeter is defined as the boundary, an annulus requires us to add the circumference of both concentric circles:

Perimeter of a polygon formula (regular pentagon, hexagon, octagon, etc.)

In our perimeter calculator, we've also implemented a simple formula for a regular polygon perimeter:

where n n n is the number of polygon sides. So, for example, you can calculate the perimeter of a pentagon, hexagon, or octagon.

Additionally, for polygons up to 12 sides, the polygon name will appear in the tool. Awesome!

If you want to determine the perimeter of any polygon, sum the lengths of all its sides:

where a 1 a_1 a 1 ​ , a 2 a_2 a 2 ​ , ..., a n a_n a n ​ are sides lengths, and ∑ \sum ∑ is the sum symbol (from i = 1 i = 1 i = 1 to n n n ).

Or use the vertices coordinates:

With x n + 1 = x 1 x_{n+1}=x_1 x n + 1 ​ = x 1 ​ and y n + 1 = y 1 y_{n+1}=y_1 y n + 1 ​ = y 1 ​ .

How do I calculate the perimeter of irregular shapes?

To find the perimeter of an irregular figure:

• Measure the lengths of all (outer) sides.
• If the sides include circular fragments , measure the radius and the central angle, i.e., the angle between the radii that join the two endpoints of the arc to the center.
• Apply the circle circumference formula for this radius and take the part proportional to the angle.
• Add together the length of all sides.

Can I determine area given perimeter?

In general, no , it's not possible to calculate area from the perimeter. This is particularly true for rectangles, parallelograms, kites, and trapezoids. However , for some specific shapes, like squares, hexagons, regular polygons in general, and circles, you can determine their side (radius in the case of circles) from the perimeter and then proceed to compute the area.

What is the perimeter of a 20m by 15m rectangular building?

The perimeter is 70 m . To arrive at this result, you need to add together the length of all four sides of the building. Two sides of length 20 m added together give 40 m, while the other two sides of length 15 m added together give 30 m. Together, we get 40 m + 30 m = 70 m, as claimed.

Circle circumference

• Math Article

The perimeter of a two-dimensional figure is the distance covered around it. It defines the length of shape, whether it is a triangle, square, rectangle or a circle . Area and perimeter are the two major properties of a 2D shape , which describes them.

The perimeter of each shape varies as per their dimensions. Only in the case of a circle, the perimeter is stated as the circumference of the circle . But the method to find the perimeter of all the polygons is the same, which is we need to add all its sides.

If we need to calculate the length of a circular or rectangular field, then with the help of the perimeter formula we can easily find it, given the dimensions. Let us learn the formula here to find the perimeter for all the two-dimensional shapes.

Also, read:

• Area and Perimeter Formula
• Perimeter Of Shapes
• Perimeter Of Polygons

Perimeter Meaning

The perimeter of any two-dimensional closed shape is the total distance around it. Perimeter is the sum of all the sides of a polygon, such as:

• Perimeter of square = Sum of all four sides
• Perimeter of rectangle = Sum of all four sides
• Perimeter of triangle = Sum of all three sides

Here is the list of formulas of the perimeter for all the 2d-shapes.

How to Find Perimeter

There are different ways to find the perimeter of a given shape apart from the formulas given above. We can use a ruler to measure the length of the sides of a small regular shape such as square, rectangle, parallelogram, etc. The perimeter will be obtained by adding the measurements of the sides/edges of the shape. 

We can use a string or thread for small irregular shapes. In this case, place either a string or thread precisely along the figure’s boundary once. The total length of the string used along the border of the shape is its perimeter.

Perimeter Units

Units are essential when representing the parameters of any geometric figure. For example, the length of a line segment measured is 10 cm or 10 m, here cm and m represent the units of measurement of the length. Similarly, the units for perimeter are the same as for the length of the sides or given parameter. If the length of the side of a square is given cm, then the units for perimeter will be in cm. There is another case, where the dimensions are given in two different units such as length of a rectangle in ft and breadth in inches, then units for the perimeter of a rectangle will be ft, for this we need to convert both the measurements into ft.

We will solve here some of the example questions to understand how to find the perimeter of different shapes.

Question.1: What is the perimeter of an equilateral triangle whose side length is 7 cm?

Solution: Given, the length of the side of an equilateral triangle is 7 cm

As we know, the equilateral triangle has all its sides equal in length.

Perimeter of triangle = a+b+c

Perimeter = 3a

P = 3 x 7 = 21 cm

Question 2: If the length of parallel sides of a parallelogram is 8 cm and 11 cm, respectively, then find its perimeter.

The length of parallel sides of a parallelogram is 8 cm and 11 cm, respectively.

By the formula of perimeter, we know;

Perimeter of Parallelogram = 2(a+b)

P = 2 (8 + 11)

Therefore, the perimeter of a given parallelogram is 38 cm.

Question 3: If the radius of a circle is 21 cm, then find its perimeter.

Solution: Given,

Radius of circle = 21 cm

Perimeter of circle = Circumference of circle = 2πr

Circumference = 2 × 22/7 × 21

= 2 × 22 × 3

Therefore, the perimeter of the circle here is equal to 21 cm.

Question 4: A regular pentagon of side 3 cm is given. Find its perimeter.

Solution: Given, the length of the side of a regular pentagon = 3 cm

As we know, a regular pentagon will have all its 5 sides equal.

The perimeter of the regular pentagon = 5a, where a is the side length

Perimeter = 5 × 3

Therefore, the answer is 15 cm here.

Put your understanding of this concept to test by answering a few MCQs. Click ‘Start Quiz’ to begin!

Select the correct answer and click on the “Finish” button Check your score and answers at the end of the quiz

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Word Problems: Area and Perimeter of a Rectangle

In geometry, calculating the area and perimeter of rectangles is typically one of the first lessons we learn. While some of us may find this concept easy, it's worth noting that these skills have real-world applications. Math is not solely for theoretical problem-solving on paper; it can also help us gain valuable insights about the world around us. We will explore how to solve word problems related to finding the area and perimeter of a rectangle.

Review: What is a rectangle?

As you might recall, a rectangle is a parallelogram with four right angles. While a rectangle falls into the general category of a parallelogram, not all parallelograms are rectangles. Interestingly enough, squares are also considered rectangles -- but not all rectangles are squares. This is what a typical rectangle looks like:

Review: Finding the perimeter and area of a rectangle

Let's quickly review how to find the perimeter of a rectangle.

As we may recall, the formula for finding the perimeter is quite simple: P = 2 l + 2 w where P is the perimeter, l is the length, and w is the width.

We may also recall that the formula for area is A = l ⁢ w

Things really start to get tricky when we are only given certain values, and we need to find the missing values. These questions can be more difficult than they seem at first.

Solving word problems

Now we're ready to start solving some word problems with our knowledge of rectangle perimeter and area:

Let's assume that we have a swimming pool with a perimeter of 56 meters. We also know that the length of the pool is 16 meters. Can we use these values to determine the width of our pool?

Let's start by visualizing the problem:

Next, let's remind ourselves of the formula for perimeter: P = 2l+2w

Now let's plug in our known values: f o r m u l a = 2 ( 1 6 ) + 2 w

Now we can simplify: formula56 = 32 + 2 w

Next, we can simplify even further by subtracting 32 from both sides: f = a ⁢ x ⁢ b c ⁢ x ⁢ d

Now all we need to do is ask ourselves what value equals 24 when multiplied by two. In other words, we need to divide both sides of the equation by 2. We are left with: w = 12

The width of this pool must be 12 meters.

Let's try another question:

Let's say we have a rectangular fence. We know that this fence encloses an area of 500 square feet. We also know that the width of this fenced enclosure is 20 feet. Using these values, can we determine the length of the fence?

First, let's visualize the problem:

Next, let's remind ourselves of the formula for the area of a rectangle: A = l ⁢ w

Now all that we need to do is divide each side of the equation by 20 to isolate and determine the length f = 25 .

In other words, the length of our rectangular fenced enclosure is 25 feet.

Topics related to the Word Problems: Area and Perimeter of a Rectangle

Word Problems

Flashcards covering the Word Problems: Area and Perimeter of a Rectangle

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Common Core: 4th Grade Math Flashcards

Practice tests covering the Word Problems: Area and Perimeter of a Rectangle

Common Core: 4th Grade Math Diagnostic Tests

4th Grade Math Practice Tests

Pair your student with a tutor who can explain word problems involving rectangular areas and perimeters

It can be difficult for students to fully grasp concepts like rectangular areas and perimeters. This is especially true when dealing with word problems, as some students process visual information better than verbal information, and vice versa. One of the key benefits of tutoring is the fact that these educators can present information in a way that is conducive to your student's learning style. For example, visual learners can draw out the problem, while verbal learners can talk the problem through with their tutors. Tutors can also personalize your student's learning experience in a number of different ways based on their ability level and even their hobbies. Each tutor is carefully vetted before working with students, ensuring a high level of academic knowledge and teaching skill. Reach out to Varsity Tutors today, and we'll pair your student with an appropriate tutor.

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REAL WORLD PROBLEMS INVOLVING AREA AND PERIMETER

Problem 1 :

The diagram shows the shape and  dimensions of Teresa’s rose garden.

(a) Find the area of the garden.

(b) Teresa wants to buy mulch for her garden. One bag of mulch covers  12 square feet. How many bags will she need?

By drawing a horizontal line, we can divide the given shape into two parts as shown below.

(1) ABCD is a rectangle

(2) CEFG is also a rectangle

Area of the garden

= Area of rectangle ABCD + Area of the rectangle CEFG

Area of rectangle ABCD :

length AB = 15 ft and width BD = 9 ft

= length x width 

 =  15 x 9

=  135 ft ²  ----(1)

Area of rectangle CEFG :

length CE = 24 ft and width CF = AF - AC ==> 18 - 9 = 9 ft

 =  24 x 9

=  216 ft ²  ----(1)

(1) + (2) 

Area of the rose garden = 135 + 216 ==> 351 ft ²

Number of bags that she needed = 351/12 ==> 29.25

So, she will need 30 bags of mulch

Problem 2 :

The length of a rectangle is 4 less than 3 times its width. If its length is 11 cm, then find the perimeter. 

Let w be the width of the rectangle.

Then, its length is (3w - 4).

Given : Length is 11 cm. 

Then, 

Length (l)  =  11

3w - 4  =  11

3w  =  15

w  =  5

So, the perimeter of the rectangle is 

=  2(l + w)

=  2(11 + 5)

=  2(16)

=  32 cm

Problem 3 :

The diagram shows the floor plan of a hotel  lobby. Carpet costs $3 per square foot. How  much will it cost to carpet the lobby?

By observing the above picture, we can find two trapeziums of same size. Since both are having same size. We can find area of one trapezium and multiply the area by 2.

Area of trapezium = (1/2) h (a +  b)

h = 15.5 ft  a = 30 ft  and b = 42 ft

  =  (1/2) x 15.5 x (30 + 42)

  =  (1/2) x 15.5 x 72 ==> 15.5 x 36==> 558 square feet

Area of floor of a hotel  lobby = 2 x 558

 =  1116 square feet

Cost of carper per square feet = $3

=  3 x 1116 ==> $ 3348

Amount spent for carpet =    $ 3348.

Problem 4 :

The cost of fencing a circle shaped garden is $20 per foot. If the radius of the garden is 14 feet, find the total cost of fencing the garden. ( π  =  22/7).  

To know the length of fencing required, find the circumference of the circle shaped garden.

Circumference of the circle shaped garden is 

=  2πr

Substitute 22/7 for π and 14 for r. 

=  2(22/7)(14)

=  88 feet

Total cost of fencing is 

=  88(20)

=  $1760

Problem 5 :

Jess is painting a giant arrow on a playground. Find the area of  the giant arrow. If one can of paint covers 100 square feet,  how many cans should Jess buy?

Now we are going to divide this into three shapes. Two triangles and one rectangle.

Area of rectangle = length x width

   =  18 x 10 ==> 180 square feet

Area of one triangle = (1/2) x b x h

   =  (1/2) x 6 x 10 ==> 30 square feet

Area of two triangles = 2 x 30 = 60 square feet

Total area of the given shape = 180 + 60

=   240 square feet

one can of paint covers 100 square feet

Number of cans needed = 240/100 = 2.4 approximately 3.

So, Jessy has to 3 cans of paint.

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Area and perimeter

What can you say about these two shapes?

What can you say about the shapes below?

Can you draw a shape in which the area is numerically equal to its perimeter? And another? Can you draw a shape in which the perimeter is numerically twice the area? Can you draw a shape in which the area is numerically twice the perimeter? Can you make the area of your shape go up but the perimeter go down? Can you make the perimeter of your shape go up but the area go down?    Can you draw some shapes that have the same area but different perimeters? Can you draw some shapes that have the same perimeter but different areas?  

Matthew from Parkgate Primary School focused on the first two shapes in the problem. He said:

Thomas from Colet Court examined the eight shapes which were drawn on the cards.  He labelled the shapes A, B, C, D, E, F, G and H, going from left to right in the top row, then left to right in the bottom row.

Thomas didn't give any units in his solution.  I guess we could say the perimeter is measured in 'units' and the area in 'square units', although some of you, like Matthew above, assumed the squares were 1cm long.  So, that would mean the perimeter is in cm and the area in cm$^2$. Thomas remarked:

Noor from Kingsbury Green Primary School answered the question; 'Can you draw a shape in which the area is numerically equal to its perimeter? And another?'.  He said :

Table 3 (they didn't give their school) looked at finding a shape which has a perimeter numerically twice the area.  They wrote:

Bashayer from Kingsbury Green Primary also found this solution.  Miiti from Kingsbury Green created this shape which has a perimeter of $20$ units and an area of $10$ square units:

Thomas from Colet Court drew a shape in which the area is numerically twice the perimeter:

Thomas went on to investigate how to make the area of a shape go up but the perimeter go down.  He said:

Thomas  also said that you can make the perimeter of a shape go up but the area go down

Joe and Charlie from Coniston Primary described the way they worked on this part of the problem:

Very, very well done all of you.  You have obviously put a lot of thought into this problem.

Area and Perimeter

Why do this problem, possible approach, key questions, possible extension, possible support.