Sphere Surface Area Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Sphere Surface Area.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The total area covering the curved outer surface of a sphere, given by the formula

S = 4\pi r^2

The 'skin area' of a perfectly round ball—the amount of material needed to cover it with no overlaps.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A sphere's outer surface area is $4\pi r^2$ — four times the area of its great-circle cross-section.

Common stuck point: The procedure for sphere surface area is the easy part; the trap is using $\pi r^2$ instead of $4\pi r^2$ . Asking "Am I covering the curved outside of a 3D ball (area in square units), not a flat circle or the inside?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I covering the curved outside of a 3D ball (area in square units), not a flat circle or the inside?

Worked Examples

Example 1

easy

Find the surface area of a sphere with radius

7

cm.

Sphere with radius 7 cm — find its surface area

Answer

SA = 196\pi \approx 615.75

cm²

First step

Step 1: Recall the surface area formula for a sphere:

SA = 4\pi r^2

Full solution

2
Step 2: Substitute $r = 7$ cm: $SA = 4\pi(7)^2 = 4\pi(49) = 196\pi$ .
3
Step 3: Calculate the numerical value: $SA = 196\pi \approx 615.75$ cm².

The surface area formula

4\pi r^2

gives the total area of the curved surface of a sphere. With radius 7 cm, we compute

4 \times \pi \times 49 = 196\pi \approx 615.75

cm².

Example 2

medium

A sphere has a surface area of

100\pi

cm². Find its radius and volume.

Example 3

medium

Find the surface area of a sphere with diameter 10 cm.

Sphere with diameter 10 cm (radius 5 cm) — find its surface area

Example 4

easy

A globe has radius

20

cm. Approximate its surface area using

\pi \approx 3.14

Example 5

medium

If a sphere's radius is tripled, by what factor does its surface area change?

Example 6

medium

Spherical bearing has surface area

16\pi

^2

. What is its diameter?

Sphere with surface area 16π mm² — find its diameter

Example 7

hard

Two metal spheres of radii

2

and

3

are melted and reformed into a single sphere. Find the surface area of the new sphere in terms of

\pi

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

A basketball has a diameter of

24

cm. Find its surface area.

Example 2

hard

Two spheres have radii in the ratio

2:3

. Find the ratio of their surface areas. If the smaller sphere has a surface area of

64\pi

cm², find the surface area of the larger sphere.

Example 3

easy

Find the surface area of a sphere with radius

1

in terms of

\pi

Example 4

easy

Find the surface area of a sphere with radius

4

in terms of

\pi

Example 5

easy

A sphere has surface area

36\pi

. Find its radius.

Example 6

easy

Find the curved surface area of a hemisphere with radius

8

, in terms of

\pi

Example 7

medium

A sphere has surface area

144\pi

. Find its volume in terms of

\pi

Example 8

medium

Two spheres have radii

r

and

3r

. Find the ratio of their surface areas.

Example 9

medium

A sphere of radius

r

has surface area equal to its circumference times what length?

Example 10

medium

A spherical scoop of ice cream of radius

3

cm sits in a cone. Find the ice cream's surface area in terms of

\pi

Example 11

medium

A sphere has surface area

S

. Express its radius in terms of

S

Example 12

medium

A balloon's radius grows from

4

cm to

6

cm. By what factor does its surface area grow?

Example 13

hard

A sphere is inscribed in a cylinder of equal height and radius. Find the ratio of the sphere's surface area to the cylinder's lateral surface area.

Example 14

hard

A hemispherical bowl of radius

5

is carved into the top face of a cube of side

10

. Find the painted outer surface area of the resulting solid (use

\pi \approx 3.14

Example 15

hard

A sphere has surface area

S

and volume

V

. Show that

S^3 = 36\pi V^2

Example 16

hard

A spherical tank has surface area

400\pi

^2

. How many gallons does it hold? (1 ft

^3

\approx 7.48

gal.)

Example 17

hard

If a sphere's radius increases by

50\%

, by what percent does its surface area increase?

Example 18

challenge

A sphere is inscribed in a cone of base radius

6

and height

8

(slant

= 10

). Find the inscribed sphere's surface area in terms of

\pi

Example 19

challenge

A spherical balloon's surface area grows at

10

^2

/s. When

r = 5

cm, how fast is

r

growing? (Use calculus.)

← Back to Sphere Surface Area Formula Explained Practice Mode

Related Concepts

Surface Area Circles Volume of a Sphere

Background Knowledge

These ideas may be useful before you work through the harder examples.

surface areacirclesvolume of sphere