problem solving surface area of sphere

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Surface Area of Sphere | Formula, Derivation and Solved Examples

A sphere is a three-dimensional object with all points on its surface equidistant from its center, giving it a perfectly round shape. The surface area of a sphere is the total area that covers its outer surface.

To calculate the surface area of a sphere with radius r, we use the formula:

Surface Area of Sphere = 4πr 2

This formula shows that the surface area of a sphere is directly proportional to the square of its radius. For example, if the radius of a sphere is doubled, its surface area increases by a factor of four. Let’s learn more about the formula for the Surface Area of the Sphere with solved examples.

Table of Content

What is Surface Area of Sphere?

Surface area of sphere formula, derivation of surface area of sphere, how to find surface area of sphere, curved surface area (csa) of sphere.

Total Surface Area (TSA) of Sphere

Solved Examples on Surface Area of Sphere

Practice problems – surface area of sphere.

The surface area of a sphere is the region covered by the outer surface in the 3-dimensional space. It can be said that a sphere is the 3-dimensional form of a circle. The surface area of a sphere formula is given in terms of pi (π) and radius.

Formula for surface area of a sphere:

Surface Area of Sphere Formula using diameter:

Surface Area of Sphere = πd 2

Where, r is the radius of the sphere.

We know that a sphere is round in shape, so to calculate its surface area, we can connect it to a curved shape, such as a cylinder. A cylinder is a three-dimensional figure that has a curved surface with two flat surfaces on either side.

Let’s consider that the radius of a sphere and the radius of a cylinder to be the same, so the sphere can perfectly fit into the cylinder.

Therefore, the height of the sphere = height of a cylinder = the diameter of a sphere.

Surface area of a sphere = Lateral surface area of a cylinder

We know that,

The lateral surface area of a cylinder = 2πrh,

Where r is the radius of the cylinder and h is its height.

We have assumed that the sphere perfectly fits into the cylinder. So, the height of the cylinder is equal to the diameter of the sphere.

Height of the cylinder (h) = Diameter of the sphere (d) = 2r (where r is the radius)

The Surface area of a sphere = The Lateral surface area of a cylinder = 2πrh Surface area of the sphere = 2πr × (2r) = 4πr 2

Hence , the surface area of the sphere = 4πr 2 square units

Total Surface Area ( TSA ) of Sphere

As the complete surface of the sphere is curved thus total Surface Area and Curved Surface Area are the same for the Sphere.

TSA of Sphere = CSA of Sphere

Sphere: Definition and Properties Sphere Formulas Radius Formula pi (π) Formula

Let’s solve questions on the Surface Area of Sphere.

Example 1: Calculate the total surface area of a sphere with a radius of 15 cm. (Take π = 3.14 )

Given, the radius of the sphere = 15 cm We know that the total surface area of a sphere = 4 π r 2 square units = 4 × (3.14) × (15) 2 = 2826 cm 2 Hence, the total surface area of the sphere is 2826 cm 2 .

Example 2: Calculate the diameter of a sphere whose surface area is 616 square inches. (Take π = 22/7)

Given, the curved surface area of the sphere = 616 sq. in We know, The total surface area of a sphere = 4 π r 2 square units ⇒ 4 π r 2 = 616 ⇒ 4 × (22/7) × r 2 = 616 ⇒ r 2 = (616 × 7)/(4 × 22) = 49 ⇒ r = √49 = 7 in We know, diameter = 2 × radius = 2 × 7 = 14 inches Hence, the diameter of the sphere is 14 inches.

Example 3: Find the cost required to paint a ball that is in the shape of a sphere with a radius of 10 cm. The painting cost of the ball is ₨ 4 per square cm. (Take π = 3.14)

Given, the radius of the ball = 10 cm We know that, The surface area of a sphere = 4 π r 2 square units = 4 × (3.14) × (10) 2 = 1256 square cm Hence, the total cost to paint the ball = 4 × 1256 = ₨ 5024/-

Example 4: Find the surface area of a sphere whose diameter is 21 cm. (Take π = 22/7)

Given, the diameter of a sphere is 21 cm We know, diameter = 2 × radius ⇒ 21 = 2 × r ⇒ r = 10.5 cm Now, the surface area of a sphere = 4 π r 2 square units = 4 × (22/7) × (10.5) = 1386 sq. cm Hence, the total surface area of the sphere = 1386 sq. cm.

Problem 1: Find the surface area of a sphere with a radius of 5 cm. Use π=3.14.

Problem 2: A sphere has a diameter of 10 inches. Calculate its surface area.

Problem 3: Determine the surface area of a sphere whose radius is 7 meters.

Problem 4: The radius of a sphere is 15 cm. What is its surface area in square centimeters?

Problem 5: If a sphere with a radius of 8 cm is cut into two hemispheres, what is the surface area of each hemisphere?

Problem 6: A sphere has a surface area of 500 square meters. What is the radius of the sphere?

Problem 7: Calculate the surface area of a sphere with a radius of 12 cm.

Problem 8: The diameter of a sphere is 16 inches. Find its surface area.

Problem 9: A spherical balloon has a radius of 21 cm. Determine its surface area in square centimeters.

Problem 10: A sphere has a radius of 0.5 meters. Find its surface area in square meters.

FAQs on Surface Area of Sphere

What is surface area of sphere .

The surface area of a sphere is given by the formula, Surface area = 4πr 2 Where r is the radius of the sphere.

What is Surface Area of Hemisphere?

The surface area of a hemisphere is given by the sum of half of the sphere’s surface area and the base area (which is circular), that is, S.A. = 2πr 2 + πr 2 = 3πr 2 Thus, the surface area of a hemisphere = 3πr 2

What is Lateral Surface Area of Sphere?

The lateral surface area is equal to the surface area of the sphere, which is equal to the curved surface area of the sphere.

Why is Surface Area of Sphere 4 Times the Area of Circle?

Surface area of a sphere is four times the area of a circle because a sphere with radius, r can be thought of as composed of circles with the same radius. The formula for the surface area of a sphere, 4πr 2 is derived from the fact that its surface area is equivalent to the lateral surface area of a cylinder with the same radius and height equal to the diameter of the sphere. The diameter is twice the radius, thus leading to the factor of four.

How Does Surface Area of Sphere Change When Radius is Halved?

When the radius of a sphere is halved, its surface area becomes one-fourth of the original surface area. This is because the surface area formula, 4πr 2 , involves the square of the radius, so halving the radius squares one-fourth.

How does Surface Area of Sphere Change When Radius is Tripled?

When the radius of a sphere is tripled, its surface area increases by a factor of nine. This is because the surface area is proportional to the square of the radius (4πr 2 ) , so tripling the radius (3 r ) leads to 4π(3r) 2 = 9×4πr 2 , which is nine times the original surface area.

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Surface area of a sphere

Here you will learn about the surface area of a sphere, including how to calculate the surface area of a sphere given its radius and the surface area of a hemisphere.

Students will first learn about the surface area of a sphere as a part of geometry in 8 th grade.

What is the surface area of a sphere?

The surface area of a sphere is the area which covers the outer surface of the sphere. It includes both the curved surface area of a sphere and the area of the sphere’s circular base, also known as the total surface area of a sphere ( tsa ).

The formula for the surface area of a sphere with radius r is:

\text{Surface area}=4\pi{r}^{2}

Notice that the square of the radius (r^2) occurs within the surface area formula.

The surface area of a shape uses two-dimensions , so the units for surface area are units squared.

Common Core State Standards

How does this relate to 8 th grade math?

Grade 8: Geometry (8.G.C.9) Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

[FREE] Surface Area Worksheet (Grade 6)

Use this quiz to check your grade 6 students’ understanding of surface area. 10+ questions with answers covering a range of 6th grade surface area topics to identify areas of strength and support!

How to calculate the surface area of a sphere

In order to calculate the surface area of a sphere:

Write down the formula.

Substitute the given values into the formula.

Complete the calculation.

Write the final answer, including the units.

Surface area of a sphere examples

Example 1: surface area of a sphere given the radius.

Find the surface area of the sphere below. Write your answer to 1 decimal place.

To answer the question, use the formula for the surface area of a sphere:

2 Substitute the given values into the formula.

The radius (the distance from the center of the sphere to the surface) is 4.8 \, cm.

Substitute the value of the radius into the formula for the surface area of a sphere:

3 Complete the calculation.

Use a calculator to find the value for the surface area:

4 Write the final answer, including the units.

Check what form the final answer needs to be. Here you are asked to give the answer to 1 decimal place.

The surface area of the sphere is 289.5 \mathrm{~cm}^2 \, \text{(1dp)}.

Example 2: surface area of a sphere

Find the surface area of a sphere with the diameter of 42 \mathrm{~mm}.

Write your answer to the nearest square millimeter.

\text{Surface area}=4 \pi r^2

Because the diameter is twice the length of the radius, divide the diameter by 2 to get the value for r \text{:}

\begin{aligned}r&=d\div{2} \\\\ &=42\div{2} \\\\ &=21\end{aligned}

So the radius r=21 \, mm. Now, substitute r=21 into the formula for the surface area of a sphere to get:

\text{SA }=4 \times \pi \times 21^2

Use a calculator to work out the value.

\begin{aligned}\text{SA }&=4 \times \pi \times 21^2 \\\\ &=1764\pi \\\\ &=5541.769441… \end{aligned}

Here you are asked to write the answer to the nearest square millimeter and so:

5541.769441… = 5542 \, \text{mm}^2 \, \text{(0dp)}

The surface area of the sphere is 5542 \mathrm{~mm}^2 \, \text{(0dp)}.

Example 3: surface area of a sphere – in terms of π

Find the surface area of a sphere with a radius of 6 \, m.

Leave your answer in terms of \pi.

Substitute the value of the radius into the surface area of a sphere formula.

\text{SA }=4 \times \pi \times 6^2

Work out the curved surface area, focussing on the number parts of the calculation.

Here you are asked to give the answer in terms of \pi.

144\pi\text{ m}^2

The surface area of the sphere is 144\pi\text{ m}^2.

Example 4: surface area of a sphere

Find the surface area of the sphere below. Write your answer to 2 decimal places.

The radius (the distance from the center of the sphere to the surface) is 12.4 \, cm.

\text{SA }=4 \times \pi \times 12.4^2

\begin{aligned}\text{SA }&=4 \times \pi \times 12.4^2 \\\\ & =615.04 \pi \\\\ & =1932.20515 \ldots\end{aligned}

Check what form the final answer needs to be. Here you are asked to give the answer to 2 decimal places.

1932.20515 . \ldots=1932.21 \mathrm{~cm}^2 \, \text{(2dp)}

The surface area of the sphere is 1932.21 \mathrm{~cm}^2 \, \text{(2dp)}.

The surface area of a sphere is 600 \mathrm{~cm}^2. Calculate the radius of the sphere. Write your answer to 1 decimal place.

The surface area is given, so substitute this into the formula.

600=4 \pi r^2

Rearrange the formula to find the value of r.

The radius of the sphere is 6.9 \mathrm{~cm} \, \text{(1dp)}.

The surface area of a sphere is 4000 \mathrm{~cm}^2. Calculate the radius of the sphere. Write your answer to 1 decimal place.

4000=4 \pi r^2

The radius of the sphere is 17.8 \mathrm{~cm} \, \text{(1dp)} .

Teaching tips for surface area of a sphere

Use visuals, such as diagrams and real life examples, to help students visualize what the shape of a sphere looks like and what it means to calculate the surface area.
Provide practice problems at various difficulty levels for students to work through. This can be done with worksheets, or with interactive technology. Allow students to work together and independently to reinforce understanding.
Have students find the surface area of multiple shapes and compare them. This will better help students understand the formulas they use, and why certain shapes are used in different real life situations.

Easy mistakes to make

Rounding too soon It is important to not round the answer until the end of the calculation. This will mean your final answer is accurate.
Not using the correct units For area, use square units such as \mathrm{~cm}^2. For volume, use cube units such as \mathrm{~cm}^3.
Incorrectly using the radius or the diameter It is a common error to mix up radius and diameter. Remember the radius is half of the diameter.
Confusing the volume formula with the surface area formula It is a common error to mix up the formula for the volume of a sphere with the formula for the surface area of a sphere. Remember that the surface area formula includes a squared term, whereas the volume formula includes a cubed term.

Related surface area lessons

Surface area of a prism
Surface area of a rectangular prism
Surface area of a triangular prism
Surface area of a pyramid
Surface area of a cone
Surface area of a cylinder
Surface area of a cube
Surface area of a hemisphere

Practice surface area of a sphere questions

1. Find the surface area of a sphere of radius 10 \, cm. Give your answer to 1 decimal place.

You are finding the surface area of a sphere, so substitute the value of r into the formula.

2. Find the surface area of a sphere of diameter 6.4 \, cm. Write your answer to 1 \, dp.

To find the surface area of a sphere, substitute the value of r into the formula. The diameter is given, so half 6.4 to get the radius, then substitute this value into the formula for the surface area of a sphere.

3. Calculate the surface area of a sphere of radius 9 \, m.

To find the surface area of a sphere, substitute the value of r into the formula.

4. Calculate the surface area of the sphere. Write your answer to 2 decimal places.

5. The surface area of a sphere is 9000 \mathrm{~cm}^2. Calculate the radius of the sphere. Write your answer correct to 3 significant figures.

Using the formula, we substitute the value of the surface area and rearrange to find the radius.

6. A spherical ball has a radius of 30 \, cm.

A tin of paint covers 2000 \mathrm{~in}^2.

A tin of paint costs \$ 13.

Calculate how much it would cost to paint the outside surface of the spherical ball.

To find the surface area of a sphere, substitute the value of r, the radius of the ball, into the formula.

Next, divide the surface area by the number of square inches that a tin of paint covers.

Multiply the number of tins needed by the total cost of each tin of paint.

Surface area of a sphere FAQS

A sphere is a three-dimensional shape that is perfectly round. It can be defined as the set of all points in three dimensional space that are a fixed distance (radius) from a given point (the center).

The diameter of a sphere is twice the length of its radius. If you are given the radius of the sphere, simply double it to find the diameter. Alternatively, if you are given the diameter and need the radius, you can divide the diameter by 2 to find the radius.

A circle is a two-dimensional shape with a flat surface. A sphere is a three dimensional shape that is perfectly round, like a ball. The circle can also be thought of as the cross section of a sphere.

Did you know?

Archimedes was a famous ancient Greek mathematician who lived about 2,200 years ago in Sicily. He wrote about the surface area of a sphere and said it was four times the area of the greatest circle. This is the same formula that is used today.

Archimedes also worked on cylinders and discovered the formula for the lateral surface area of a cylinder. The lateral surface area excludes its circular base and top. This is usually referred to as its curved surface area.

The next lessons are

Pythagorean Theorem
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