Surface Area of a Sphere Formula

The surface area of a sphere is the total area that covers its outer surface. This surface area depends on the radius, diameter, or circumference of the sphere.

Below are all the formulas for calculating the surface area of a sphere, with detailed explanations and examples for each method.

Formulas for the Surface Area of a Sphere

Using Radius:

If the radius \( r \) of the sphere is known,

The surface area \( S \) can be calculated with the formula:

\( S = 4 \pi r^2 \)

Using Diameter:

If the diameter \( d \) of the sphere is known,

The surface area \( S \) can be calculated as:

\( S = \pi d^2 \)

Using Circumference:

If the circumference \( C \) of the sphere is known,

The surface area \( S \) can be calculated as:

\( S = \dfrac{C^2}{\pi} \)

In these formulas:

  • \( r \) is the radius of the sphere
  • \( d \) is the diameter of the sphere, where \( d = 2r \)
  • \( C \) is the circumference of the sphere, where \( C = 2\pi r \)
  • \( \pi \) (Pi) is approximately equal to 3.14159

Detailed Explanation of Each Formula

1. Formula for Surface Area Using Radius

The formula \( S = 4 \pi r^2 \) calculates the surface area by squaring the radius, multiplying by \( \pi \), and then by 4. This formula is commonly used when the radius of the sphere is known.

Example 1: Calculating Surface Area with Radius

Problem: Find the surface area of a sphere with a radius of \( r = 7 \, \text{cm} \).

Solution:

  1. Write down the formula: \( S = 4 \pi r^2 \).
  2. Substitute \( r = 7 \): \( S = 4 \pi \times 7^2 \).
  3. Calculate \( r^2 \): \( S = 4 \pi \times 49 \).
  4. Multiply: \( S = 196\pi \approx 615.75 \, \text{cm}^2 \) (using \( \pi \approx 3.14159 \)).

The surface area of the sphere is approximately \( 615.75 \, \text{cm}^2 \).

2. Formula for Surface Area Using Diameter

The formula \( S = \pi d^2 \) calculates the surface area using the diameter of the sphere. This formula is useful if only the diameter is known, as it avoids the need to divide by 2 to find the radius.

Example 2: Calculating Surface Area with Diameter

Problem: A sphere has a diameter of \( d = 12 \, \text{cm} \). Find its surface area.

Solution:

  1. Write down the formula: \( S = \pi d^2 \).
  2. Substitute \( d = 12 \): \( S = \pi \times 12^2 \).
  3. Calculate \( d^2 \): \( S = \pi \times 144 \).
  4. Multiply: \( S = 144\pi \approx 452.39 \, \text{cm}^2 \).

The surface area of the sphere is approximately \( 452.39 \, \text{cm}^2 \).

3. Formula for Surface Area Using Circumference

The formula \( S = \dfrac{C^2}{\pi} \) calculates the surface area using the circumference of the sphere. This formula is convenient if only the circumference is known, as it eliminates the need to find the radius first.

Example 3: Calculating Surface Area with Circumference

Problem: A sphere has a circumference of \( C = 20 \, \text{cm} \). Find its surface area.

Solution:

  1. Write down the formula: \( S = \dfrac{C^2}{\pi} \).
  2. Substitute \( C = 20 \): \( S = \dfrac{20^2}{\pi} \).
  3. Calculate \( C^2 \): \( S = \dfrac{400}{\pi} \).
  4. Approximate \( \pi \approx 3.14159 \): \( S \approx \dfrac{400}{3.14159} \approx 127.32 \, \text{cm}^2 \).

The surface area of the sphere is approximately \( 127.32 \, \text{cm}^2 \).

These examples demonstrate how to calculate the surface area of a sphere using different known measurements, whether it’s the radius, diameter, or circumference.