The surface area of a sphere is defined as the region covered by its outer surface in three-dimensional space. A sphere is a three-dimensional figure with a round shape, similar to a circle. The surface area of the spheres is calculated using the length of their radius or the length of their diameter. The surface area is a two-dimensional unit, so we use m², cm², or other similar units.

Here, we will learn about the formula that we can use to calculate the surface area of a sphere. In addition, we will solve some problems in which we will apply this formula.

## Formula to find the surface area of a sphere

From a visual perspective, a sphere has a three-dimensional structure that is formed by rotating a disk that is circular with respect to one of the diagonals.

Let’s consider an example in which the faces of the sphere are painted. To paint the entire surface, we have to know the total amount of paint that we will need.

Therefore, the area of each face has to be known to calculate the amount of paint needed. We define this surface as the surface area.

The formula for the surface area of a sphere is given by:

$latex A_{S}=4\pi{{r}^2}$ |

where $latex A_{s}$ represents the surface area of the sphere and *r* represents the length of the radius.

## Surface area of a sphere – Examples with answers

The formula for the surface area of spheres is used to solve the following examples. Each example has its respective solution, but it is recommended that you try to solve them yourself before looking at the answer.

**EXAMPLE 1**

What is the surface area of a sphere that has a radius of 4 m?

##### Solution

We have to use the formula for the surface area with the value $latex r=4$. Therefore, we have:

$latex A_{S}=4\pi {{r}^2}$

$latex A_{S}=4\pi {{(4)}^2}$

$latex A_{S}=4\pi (16)$

$latex A_{S}=201.1$

The surface area is 201.1 m².

**EXAMPLE 2**

A sphere has a radius of 5 m. What is its surface area?

##### Solution

We use the radius $latex r = 5$ in the formula for surface area. Therefore, we have:

$latex A_{S}=4\pi {{r}^2}$

$latex A_{S}=4\pi {{(5)}^2}$

$latex A_{S}=4\pi (25)$

$latex A_{S}=314.2$

The surface area is 314.2 m².

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**EXAMPLE 3**

If a sphere has a diameter of 12 m, what is its surface area?

##### Solution

In this case, we have the diameter of the sphere. However, we only have to divide by 2 to get the radius. Therefore, we use the formula for the surface area with the length $latex r=6$:

$latex A_{S}=4\pi {{r}^2}$

$latex A_{S}=4\pi {{(6)}^2}$

$latex A_{S}=4\pi (36)$

$latex A_{S}=452.4$

The surface area is 452.4 m².

**EXAMPLE 4**

If a sphere has a surface area of 200 m², what is its radius?

##### Solution

Here, we start with the surface area and want to find the radius. Therefore, we’ll use the surface area formula and solve for *r*:

$latex A_{S}=4\pi {{r}^2}$

$latex 200=4\pi {{r}^2}$

$latex 50=\pi {{r}^2}$

$latex 15.92={{r}^2}$

$latex r=3.99$

The length of the radius is 3.99 m.

**EXAMPLE 5**

What is the radius of a sphere that has a surface area of 460 m²?

##### Solution

We use the given value in the formula for surface area and solve for *r*:

$latex A_{S}=4\pi {{r}^2}$

$latex 460=4\pi {{r}^2}$

$latex 115=\pi {{r}^2}$

$latex 36.6={{r}^2}$

$latex r=6.05$

The length of the radius is 6.05 m.

## Surface area of a sphere – Practice problems

Use the formula for the surface area of spheres to solve the following practice problems. If you need help with this, you can look at the solved examples above.

## See also

Interested in learning more about spheres? Take a look at these pages:

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