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: Sphere surface area is 4\pi r^2—it grows with the square of the radius, so doubling radius quadruples area.

Common stuck point: Students confuse surface area 4\pi r^2 with volume \frac{4}{3}\pi r^3—note the different exponents.

Sense of Study hint: When you see a sphere problem, first identify the radius r. Then plug into S = 4\pi r^2: square the radius, multiply by \pi, then multiply by 4. Finally, check your units are squared (e.g., cm^2).

Worked Examples

Example 1

easy
Find the surface area of a sphere with radius 7 cm.

Solution

  1. 1
    Step 1: Recall the surface area formula for a sphere: SA = 4\pi r^2.
  2. 2
    Step 2: Substitute r = 7 cm: SA = 4\pi(7)^2 = 4\pi(49) = 196\pi.
  3. 3
    Step 3: Calculate the numerical value: SA = 196\pi \approx 615.75 cm².

Answer

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.

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.
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Background Knowledge

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

surface areacirclesvolume of sphere