Surface Area of a Sphere Calculator

Published on: August 4, 2024
Final Answer: Free • Full Steps: Plus

This Surface Area of a Sphere Calculator finds the total area of the outside of a sphere from its radius, and shows the working line by line. A sphere has four times the surface area of the flat circle cut through its middle, so the radius is squared first and that result is multiplied by four and by \(\pi\). The answer is written in square units and is usually rounded. Following the same pattern every time is what makes a sphere easy to check and repeat.

Sphere with radius r

In the diagram, the radius is labeled \(r\).

Step-by-step method

  1. Set up the problem.
  2. Write the formula.
  3. Substitute the radius into the formula.
  4. Square the radius.
  5. Multiply by four and by pi to get the surface area.

Formula:

\(SA = 4\pi r^{2}\)
See Example 1 Hide Example 1

Problem

\(r = 7\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the formula.
  3. Substitute the radius into the formula.
  4. Square the radius.
  5. Multiply by four and by pi to get the surface area.

Step 1

Step 1 - Set up the problem.

In this problem: We are finding the surface area of a sphere whose radius is \(r = 7\).

\(r = 7\)

Step 2

Step 2 - Write the formula.

In this problem: A sphere has four times the surface area of the flat circle through its middle, so the radius is squared and then multiplied by four and by \(\pi\).

\(SA = 4\pi r^{2}\)

Step 3

Step 3 - Substitute the radius into the formula.

In this problem: Replacing \(r\) with \(7\) in \(SA = 4\pi r^{2}\) gives \(SA = 4 \times \pi \times 7^{2}\).

\(SA = 4 \times \pi \times 7^{2}\)

Step 4

Step 4 - Square the radius.

In this problem: Squaring the radius gives \(7^{2} = 7 \times 7 = 49\).

\(SA = 4 \times \pi \times 49\)

Step 5

Step 5 - Multiply by four and by pi to get the surface area.

In this problem: Multiplying that by four and by \(\pi\) gives \(4 \times \pi \times 49 \approx 615.75\).

\(SA = 4 \times \pi \times 49 \approx 615.75\)

Final Answer

\(SA \approx 615.75\)
See Example 2 Hide Example 2

Problem

\(r = 2.5\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the formula.
  3. Substitute the radius into the formula.
  4. Square the radius.
  5. Multiply by four and by pi to get the surface area.

Step 1

Step 1 - Set up the problem.

In this problem: We are finding the surface area of a sphere whose radius is \(r = 2.5\).

\(r = 2.5\)

Step 2

Step 2 - Write the formula.

In this problem: A sphere has four times the surface area of the flat circle through its middle, so the radius is squared and then multiplied by four and by \(\pi\).

\(SA = 4\pi r^{2}\)

Step 3

Step 3 - Substitute the radius into the formula.

In this problem: Replacing \(r\) with \(2.5\) in \(SA = 4\pi r^{2}\) gives \(SA = 4 \times \pi \times 2.5^{2}\).

\(SA = 4 \times \pi \times 2.5^{2}\)

Step 4

Step 4 - Square the radius.

In this problem: Squaring the radius gives \(2.5^{2} = 2.5 \times 2.5 = 6.25\).

\(SA = 4 \times \pi \times 6.25\)

Step 5

Step 5 - Multiply by four and by pi to get the surface area.

In this problem: Multiplying that by four and by \(\pi\) gives \(4 \times \pi \times 6.25 \approx 78.54\).

\(SA = 4 \times \pi \times 6.25 \approx 78.54\)

Final Answer

\(SA \approx 78.54\)

Example Problems

Click one to load it into the box and solve it.