angles of polygons assignment

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Polygons: Formula and Examples

Exterior angles and interior angles, interior angle sum theorem, what is true about the sum of interior angles of a polygon .

The sum of the measures of the interior angles of a convex polygon with n sides is $ (n-2)\cdot180^{\circ} $

What is the total number degrees of all interior angles of a triangle ?

You can also use Interior Angle Theorem :$$ (\red 3 -2) \cdot 180^{\circ} = (1) \cdot 180^{\circ}= 180 ^{\circ} $$

What is the total number of degrees of all interior angles of the polygon ?

360° since this polygon is really just two triangles and each triangle has 180°

You can also use Interior Angle Theorem :$$ (\red 4 -2) \cdot 180^{\circ} = (2) \cdot 180^{\circ}= 360 ^{\circ} $$

What is the sum measure of the interior angles of the polygon (a pentagon) ?

Use Interior Angle Theorem :$$ (\red 5 -2) \cdot 180^{\circ} = (3) \cdot 180^{\circ}= 540 ^{\circ} $$

What is sum of the measures of the interior angles of the polygon (a hexagon) ?

Video Tutorial on Interior Angles of a Polygon

Definition of a Regular Polygon:

Examples of regular polygons.

Measure of a Single Interior Angle

What about when you just want 1 interior angle.

In order to find the measure of a single interior angle of a regular polygon  (a polygon with sides of equal length and angles of equal measure) with n sides, we calculate the sum interior angles or $$ (\red n-2) \cdot 180 $$ and then divide that sum by the number of sides or $$ \red n$$.

$ \text {any angle}^{\circ} = \frac{ (\red n -2) \cdot 180^{\circ} }{\red n} $

So, our new formula for finding the measure of an angle in a regular polygon is consistent with the rules for angles of triangles that we have known from past lessons.

To find the measure of an interior angle of a regular octagon, which has 8 sides, apply the formula above as follows: $ \text{Using our new formula} \\ \text {any angle}^{\circ} = \frac{ (\red n -2) \cdot 180^{\circ} }{\red n} \\ \frac{(\red8-2) \cdot 180}{ \red 8} = 135^{\circ} $

Finding 1 interior angle of a regular Polygon

What is the measure of 1 interior angle of a regular octagon?

Substitute 8 (an octagon has 8 sides) into the formula to find a single interior angle

Calculate the measure of 1 interior angle of a regular dodecagon (12 sided polygon)?

Substitute 12 (a dodecagon has 12 sides) into the formula to find a single interior angle

Calculate the measure of 1 interior angle of a regular hexadecagon (16 sided polygon)?

Substitute 16 (a hexadecagon has 16 sides) into the formula to find a single interior angle

Challenge Problem

What is the measure of 1 interior angle of a pentagon?

This question cannot be answered because the shape is not a regular polygon. You can only use the formula to find a single interior angle if the polygon is regular!

Consider, for instance, the ir regular pentagon below.

You can tell, just by looking at the picture, that $$ \angle A    and    \angle B $$ are not congruent .

The moral of this story- While you can use our formula to find the sum of the interior angles of any polygon (regular or not), you can not use this page's formula for a single angle measure--except when the polygon is regular .

How about the measure of an exterior angle?

Formula for sum of exterior angles: The sum of the measures of the exterior angles of a polygon, one at each vertex, is 360°.

Measure of a Single Exterior Angle

$$ \angle1 + \angle2 + \angle3 = 360° $$

$$ \angle1 + \angle2 + \angle3 + \angle4 = 360° $$

$$ \angle1 + \angle2 + \angle3 + \angle4 + \angle5 = 360° $$

Practice Problems

Calculate the measure of 1 exterior angle of a regular pentagon?

Substitute 5 (a pentagon has 5sides) into the formula to find a single exterior angle

What is the measure of 1 exterior angle of a regular decagon (10 sided polygon)?

Substitute 10 (a decagon has 10 sides) into the formula to find a single exterior angle

What is the measure of 1 exterior angle of a regular dodecagon (12 sided polygon)?

Substitute 12 (a dodecagon has 12 sides) into the formula to find a single exterior angle

What is the measure of 1 exterior angle of a pentagon?

This question cannot be answered because the shape is not a regular polygon. Although you know that sum of the exterior angles is 360 , you can only use formula to find a single exterior angle if the polygon is regular!

Consider, for instance, the pentagon pictured below. Even though we know that all the exterior angles add up to 360 °, we can see, by just looking, that each $$ \angle A \text{ and } and \angle B $$ are not congruent..

Determine Number of Sides from Angles

It's possible to figure out how many sides a polygon has based on how many degrees are in its exterior or interior angles.

If each exterior angle measures 10°, how many sides does this polygon have?

Use formula to find a single exterior angle in reverse and solve for 'n'.

If each exterior angle measures 20°, how many sides does this polygon have?

If each exterior angle measures 15°, how many sides does this polygon have?

If each exterior angle measures 80°, how many sides does this polygon have?

When you use formula to find a single exterior angle to solve for the number of sides , you get a decimal (4.5), which is impossible. Think about it: How could a polygon have 4.5 sides? A quadrilateral has 4 sides. A pentagon has 5 sides.

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Unit 2: Angles

About this unit.

In this topic, we will learn what an angle is and how to label, measure and construct them. We will also explore special types of angles.

Angle introduction

• Angles: introduction (Opens a modal)
• Naming angles (Opens a modal)
• Angle basics review (Opens a modal)
• Angle basics 4 questions Practice
• Name angles 4 questions Practice

Measuring angles

• Measuring angles in degrees (Opens a modal)
• Measuring angles using a protractor (Opens a modal)
• Measuring angles using a protractor 2 (Opens a modal)
• Measuring angles review (Opens a modal)
• Measure angles 4 questions Practice

Constructing angles

• Constructing angles (Opens a modal)
• Constructing angles review (Opens a modal)
• Draw angles 7 questions Practice

Angles in circles

• Angle measurement & circle arcs (Opens a modal)
• Angles in circles word problems (Opens a modal)
• Angles in circles 7 questions Practice

Angle types

• Recognizing angles (Opens a modal)
• Drawing acute, right and obtuse angles (Opens a modal)
• Identifying an angle (Opens a modal)
• Angle types review (Opens a modal)
• Angle types 4 questions Practice
• Recognize angles in figures 4 questions Practice
• Draw right, acute, and obtuse angles 7 questions Practice
• Benchmark angles 7 questions Practice

Vertical, complementary, and supplementary angles

• Complementary & supplementary angles (Opens a modal)
• Complementary and supplementary angles review (Opens a modal)
• Vertical angles review (Opens a modal)
• Angle relationships example (Opens a modal)
• Vertical angles are congruent proof (Opens a modal)
• Identifying supplementary, complementary, and vertical angles 7 questions Practice
• Complementary and supplementary angles (visual) 4 questions Practice
• Complementary and supplementary angles (no visual) 7 questions Practice
• Vertical angles 4 questions Practice

Angles between intersecting lines

• Angles, parallel lines, & transversals (Opens a modal)
• Parallel & perpendicular lines (Opens a modal)
• Missing angles with a transversal (Opens a modal)
• Parallel lines & corresponding angles proof (Opens a modal)
• Missing angles (CA geometry) (Opens a modal)
• Proving angles are congruent (Opens a modal)
• Proofs with transformations (Opens a modal)
• Angle relationships with parallel lines 7 questions Practice
• Line and angle proofs 4 questions Practice

Sal's old angle videos

• Intro to angles (old) (Opens a modal)
• Angles (part 2) (Opens a modal)
• Angles (part 3) (Opens a modal)
• Angles formed between transversals and parallel lines (Opens a modal)
• Angles of parallel lines 2 (Opens a modal)
• The angle game (Opens a modal)
• The angle game (part 2) (Opens a modal)
• Acute, right, & obtuse angles (Opens a modal)

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How to Understand and Master Polygons and Angles

Polygons are shapes made up of straight lines, like triangles, rectangles, and pentagons. The corners where these lines meet create angles. These angles can tell us a lot about the polygon. For instance, in a square, all angles are the same and measure \(90\) degrees. When we delve deeper into studying polygons, understanding these angles becomes key. They help in drawing, designing, and even in real-life tasks like building or sewing. In short, by exploring polygons and their angles, we uncover a mix of math, art, and everyday practicality.

Step-by-step Guide: Polygons and Angles

1. Definition of a Polygon: A polygon is a closed two-dimensional shape with straight sides. Polygons can be classified based on the number of sides they have, e.g., triangle (\(3\) sides), quadrilateral (\(4\) sides), pentagon (\(5\) sides), and so on. 2. Sum of Interior Angles: The sum of interior angles in any polygon can be determined using the formula: Sum of Interior Angles \(= (n-2) \times 180^\circ\) where \( n \) is the number of sides in the polygon. 3. Each Interior Angle of a Regular Polygon: A regular polygon has all its sides and angles equal. The measure of each interior angle can be found using: Each Interior Angle \( = \frac{\text{Sum of Interior Angles}}{n}\) where \( n \) is the number of sides.

Example 1: Find the sum of the interior angles of a pentagon.

Solution: For a pentagon, \( n = 5 \). Sum of Interior Angles \( = (5-2) \times 180^\circ = 3 \times 180^\circ = 540^\circ\)

Example 2: Calculate the sum of the interior angles of an octagon.

Solution: For an octagon, \( n = 8 \). Sum of Interior Angles \(= (8-2) \times 180^\circ = 6 \times 180^\circ = 1080^\circ\)

Example 3: Determine the measure of each interior angle of a regular hexagon.

Solution: For a hexagon, \( n = 6 \). Sum of Interior Angles \(= (6-2) \times 180^\circ = 720^\circ \). Each interior angle: \(= \frac{720^\circ}{6} = 120^\circ\)

Practice Questions:

• What is the sum of the interior angles of a triangle?
• If a regular polygon has an interior angle of \(150^\circ\), how many sides does it have?
• Determine the sum of the interior angles for a decagon (\(10\) sides).
• \(180^\circ\).
• \(12\) sides.
• \(1440^\circ\).

by: Effortless Math Team about 7 months ago (category: Articles )

Effortless Math Team

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Angles and Polygons

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• Math Article
• Interior Angles Of A Polygon

Interior Angles of a Polygon

Interior Angles of A Polygon: In Mathematics, an angle is defined as the figure formed by joining the two rays at the common endpoint. An interior angle is an angle inside a shape. The polygons are the closed shape that has sides and vertices. A regular polygon has all its interior angles equal to each other. For example, a square has all its interior angles equal to the right angle or 90 degrees. 

The interior angles of a polygon are equal to a number of sides. Angles are generally measured using degrees or radians. So, if a polygon has 4 sides, then it has four angles as well. Also, the sum of interior angles of different polygons is different. 

What is Meant by Interior Angles of a Polygon?

An interior angle of a polygon is an angle formed inside the two adjacent sides of a polygon. Or, we can say that the angle measures at the interior part of a polygon are called the interior angle of a polygon. We know that the polygon can be classified into two different types, namely:

• Regular Polygon
• Irregular Polygon

For a regular polygon, all the interior angles are of the same measure. But for irregular polygon, each interior angle may have different measurements.

Sum of Interior Angles of a Polygon

The Sum of interior angles of a polygon is always a constant value. If the polygon is regular or irregular, the sum of its interior angles remains the same. Therefore, the sum of the interior angles of the polygon is given by the formula:

Sum of the Interior Angles of a Polygon = 180 (n-2) degrees

As we know, there are different types of polygons. Therefore, the number of interior angles and the respective sum of angles is given below in the table.

Interior angles of Triangles

A triangle is a polygon that has three sides and three angles. Since, we know, there is a total of three types of triangles based on sides and angles. But the angle of the sum of all the types of interior angles is always equal to 180 degrees. For a regular triangle, each interior angle will be equal to:

180/3 = 60 degrees

60°+60°+60° = 180°

Therefore, no matter if the triangle is an acute triangle or obtuse triangle or a right triangle, the sum of all its interior angles will always be 180 degrees.

Interior Angles of Quadrilaterals

In geometry, we have come across different types of quadrilaterals, such as:

• Parallelogram

All the shapes listed above have four sides and four angles. The common property for all the above four-sided shapes is the sum of interior angles is always equal to 360 degrees. For a regular quadrilateral such as square, each interior angle will be equal to:

360/4 = 90 degrees.

90° + 90° + 90° + 90° = 360°

Since each quadrilateral is made up of two triangles, therefore the sum of interior angles of two triangles is equal to 360 degrees and hence for the quadrilateral. 

Interior angles of Pentagon

In case of the pentagon, it has five sides and also it can be formed by joining three triangles side by side. Thus, if one triangle has sum of angles equal to 180 degrees, therefore, the sum of angles of three triangles will be:

3 x 180 = 540 degrees

Thus, the angle sum of the pentagon is 540 degrees.

For a regular pentagon, each angle will be equal to:

540°/5 = 108°

108°+108°+108°+108°+108° = 540°

Interior angles of Regular Polygons

A regular polygon has all its angles equal in measure.

Interior Angle Formulas

The interior angles of a polygon always lie inside the polygon. The formula can be obtained in three ways. Let us discuss the three different formulas in detail.

If “n” is the number of sides of a polygon, then the formula is given below:

Interior angles of a Regular Polygon = [180°(n) – 360°] / n

If the exterior angle of a polygon is given, then the formula to find the interior angle is

Interior Angle of a polygon = 180° – Exterior angle of a polygon

If we know the sum of all the interior angles of a regular polygon, we can obtain the interior angle by dividing the sum by the number of sides.

Interior Angle = Sum of the interior angles of a polygon / n

“n” is the number of polygon sides.

Interior Angles Theorem

Below is the proof for the polygon interior angle sum theorem

In a polygon of ‘n’ sides, the sum of the interior angles is equal to (2n – 4) × 90°.

The sum of the interior angles = (2n – 4) right angles

ABCDE is a “n” sided polygon. Take any point O inside the polygon. Join OA, OB, OC.

For “n” sided polygon, the polygon forms “n” triangles.

We know that the sum of the angles of a triangle is equal to 180 degrees

Therefore, the sum of the angles of n triangles = n × 180°

From the above statement, we can say that

Sum of interior angles + Sum of the angles at O = 2n × 90° ——(1)

But, the sum of the angles at O = 360°

Substitute the above value in (1), we get

Sum of interior angles + 360°= 2n × 90°

So, the sum of the interior angles = (2n × 90°) – 360°

Take 90 as common, then it becomes

The sum of the interior angles = (2n – 4) × 90°

Therefore, the sum of “n” interior angles is (2n – 4) × 90°

So, each interior angle of a regular polygon is [(2n – 4) × 90°] / n

Note: In a regular polygon, all the interior angles are of the same measure.

Exterior Angles 

Exterior angles of a polygon are the angles at the vertices of the polygon, that lie outside the shape. The angles are formed by one side of the polygon and extension of the other side. The sum of an adjacent interior angle and exterior angle for any polygon is equal to 180 degrees since they form a linear pair. Also, the sum of exterior angles of a polygon is always equal to 360 degrees.

Related Articles

• Exterior Angles of a Polygon
• Exterior Angle Theorem
• Alternate Interior Angles

Solved Examples

Q.1: If each interior angle is equal to 144°, then how many sides does a regular polygon have?

Given: Each interior angle = 144°

We know that,

Interior angle + Exterior angle = 180°

Exterior angle = 180°-144°

Therefore, the exterior angle is 36°

The formula to find the number of sides of a regular polygon is as follows:

Number of Sides of a Regular Polygon = 360° / Magnitude of each exterior angle

Therefore, the number of sides = 360° / 36° = 10 sides

Hence, the polygon has 10 sides.

Q.2: What is the value of the interior angle of a regular octagon?

Solution: A regular octagon has eight sides and eight angles.

Since, we know that, the sum of interior angles of octagon, is;

Sum = (8-2) x 180° = 6 x 180° = 1080°

A regular octagon has all its interior angles equal in measure.

Therefore, measure of each interior angle = 1080°/8 = 135°.

Q.3: What is the sum of interior angles of a 10-sided polygon?

Answer: Given, 

Number of sides, n = 10

Sum of interior angles = (10 – 2) x 180° = 8 x 180° = 1440°.

Video Lesson on Angle sum and exterior angle property

Practise Questions

• Find the number of sides of a polygon, if each angle is equal to 135 degrees.
• What is the sum of interior angles of a nonagon?

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Frequently Asked Questions – FAQs

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A polygon is a plane shape with straight sides.

Is it a polygon.

Polygons are 2-dimensional shapes. They are made of straight lines, and the shape is "closed" (all the lines connect up).

Polygon comes from Greek. Poly- means "many" and -gon means "angle".

Types of Polygons

Regular or irregular.

A regular polygon has all angles equal and all sides equal, otherwise it is irregular

Concave or Convex

A convex polygon has no angles pointing inwards. More precisely, no internal angle can be more than 180°.

If any internal angle is greater than 180° then the polygon is concave . ( Think: concave has a "cave" in it )

Simple or Complex

A simple polygon has only one boundary, and it doesn't cross over itself. A complex polygon intersects itself! Many rules about polygons don't work when it is complex.

More Examples

Play with them.

Try Interactive Polygons ... make them regular, concave or complex.

Names of Polygons

You can make names using this method:

Example: a 62-sided polygon is a Hexacontadigon

BUT, for polygons with 13 or more sides, it is OK (and easier) to write " 13-gon ", " 14-gon " ... " 100-gon" , etc.

Remembering

Quadrilateral (4 sides).

Pentagon (5 Sides)

Hexagon (6 Sides)

Septagon (7 Sides)

Think Sept agon is a "Seven- agon"

Octagon (8 Sides)

Nonagon (9 Sides)

Think Non agon is a "Nine- agon"

Decagon (10 Sides)

Think Dec agon has 10 sides, just like our Dec imal system has 10 digits