# How to Find the Surface Area of Spheres?

The surface area of a sphere is the total area whose outer surface is covered. In this guide, you will learn how to find the surface area of a sphere in a few simple steps.

The formula for the surface area of a sphere depends on the radius and the diameter of the sphere. The surface area of a sphere is always expressed in square units.

**Related Topics**

**Step by step guide to** **finding the surface area of a sphere**

The area that covers the sphere’s outer surface is known as the surface of the sphere. A sphere is a three-dimensional shape of a circle. The difference between a sphere and a circle is that a circle is a \(2\)-dimensional shape, while a sphere is a \(3\)-dimensional shape. See the sphere below, which shows the center, radius, and diameter of a sphere.

**Derivation of the** **surface area of a sphere**

A sphere is round in shape, so to find its surface, we connect it in a curved shape like a cylinder. A cylinder is a shape that has a curved surface along with flat surfaces. Now, if the radius of a cylinder is the same as the radius of a sphere, it means that the sphere can fit perfectly in the cylinder. It means that the height of the cylinder is equal to the height of the sphere. Therefore, this height can also be called the diameter of the sphere. Thus, this fact was proved by a great mathematician, Archimedes, that if the radius of a cylinder and the sphere are \(r\), the surface of the sphere is equal to the lateral surface of the cylinder.

Therefore, the relationship between the surface of a sphere and the lateral surface of a cylinder is presented as follows:

Surface Area of Sphere \(=\) Lateral Surface Area of Cylinder

The lateral surface area of a cylinder \(= 2πrh\), where \(r\) is the radius and \(h\) is the height of the cylinder.

Now the height of the cylinder can also be called the diameter of the sphere because we assume that this sphere is completely fit in the cylinder. Hence, it can be said that the height of the cylinder \(=\) the diameter of the sphere \(= 2r\). Therefore, in the formula, the surface of the sphere \(= 2πrh\). \(h\) can be replaced by a diameter, that is, \(2r\). Hence, surface area of sphere is \(2πrh = 2πr(2r) = 4πr^2\).

**Formula of the surface area of a sphere**

The formula for the surface of the sphere depends on the radius of the sphere. If the radius of the sphere is \(r\) and the surface of the sphere is \(S\). Then, the surface area of the sphere is expressed as follows:

\(\color{blue}{S=4πr^2}\)

In terms of diameter, the surface area of a sphere is expressed as:

\(\color{blue}{S= 4π(\frac{d}{2})^2}\)

where \(d\) is the diameter of the sphere.

**How to calculate the surface area of a sphere?**

The surface area of a sphere is the space occupied by its surface. We can calculate the sphere surface using the sphere surface area formula. The steps we can use to calculate the surface area of a sphere are:

Let’s give an example to learn how to calculate the surface area of a sphere using its formula.

**Example:** Find the surface area of a spherical ball that has a radius of \(8\) inches.

**Solution:**

**Step 1:**Note the radius of the sphere. Here, the radius of the ball is \(8\) inches.**Step 2:**As we know, the surface area of sphere \(= 4πr^2\), so after replacing the value of \(r = 8\), we get, surface area of sphere \(= 4πr^2 = 4 × 3.14 × 8^2 = 4 × 3.14 × 64 = 803.84\)**Step 3:**Therefore, the surface area of the sphere is \(803.84\space in^2\)

**Finding the Surface Area of Spheres – Example 1:**

If the radius of a sphere is \(25\) \(cm\), find its surface area.

**Solution:**

Given, that the radius \(r\) of the sphere \(= 25\) \(cm\).

The surface area of the sphere \(= 4\pi r^2=\:4×3.14×25^2=4×3.14×625= 7,850\space cm^2\)

**Exercises for** **Finding the Surface Area of Spheres **

**Find the surface area of each sphere.** **\((π=3.14)\)**

- \(\color{blue}{ r= 4.5\space ft}\)
- \(\color{blue}{d=13 \space in}\)
- \(\color{blue}{r=22 \space cm}\)
- \(\color{blue}{d=39 \space ft}\)

- \(\color{blue}{254.34\space ft^2}\)
- \(\color{blue}{530.66 \space in^2}\)
- \(\color{blue}{6,079.04 \space cm^2}\)
- \(\color{blue}{4,775.94 \space ft^2}\)

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