Surface Area of a Sphere

In this tutorial, we'll learn how to find the surface area of a sphere. And given how often we see the spherical shape around us, I am sure we all have some intuitive understanding of what a sphere is.

So, a spherical surface is the set of all points in space that are at a fixed distance (radius, usually denoted by $\hspace{0.2em} r \hspace{0.2em}$ ) from a given point (center, $\hspace{0.2em} O \hspace{0.2em}$ ).

It’s very similar to the concept of a circle. The key difference is – a circle is the set of all points in a plane (instead of space) that are equidistant from a given point.

Surface Area of a Sphere

Unlike a cone, cube, or cylinder, a sphere does not have any edges. So a sphere has only one continuous surface. And the area covered by this surface is the surface area of the sphere.

Formula | Surface Area of a Sphere

For a sphere with a radius $\hspace{0.2em} r \hspace{0.2em}$ , the surface area is given by

A = 4 \pi r^2

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Derivation

Archimedes, the famous Greek polymath, found that the surface area of a sphere is equal to the curved surface area of a cylinder with a radius equal to the sphere's radius and height equal to the sphere's diameter.

Now, the curved surface area of a cylinder is given by –

A_c = 2 \pi r h

Replacing with radius $\hspace{0.2em} r \hspace{0.2em}$ and height $\hspace{0.2em} 2r \hspace{0.2em}$ is given by

\begin{align*} A_c &= 2 \pi r \cdot 2r \\[1em] &= 4 \pi r^2 \end{align*}

Hence the curved surface area of a sphere must be equal to

A_c = 4 \pi r^2

How to Find the Surface Area of a Sphere | Examples

Alright. Time to solve a few examples using what we have learned so far.

Example

Find the surface area of a sphere with a diameter of $\hspace{0.2em} 28$ inches.

Solution

The surface area of a sphere is given by -

A = 4 \pi r^2

But the question doesn't give us the radius $\hspace{0.2em} (r)$ . Instead, it tells us the diameter is $\hspace{0.2em} 28 \hspace{0.2em}$ inches. So first we need to get the radius.

\begin{align*} r &= \frac{d}{2} \\[1.3em] &= \frac{28}{2} = 14 \end{align*}

Now, substituting the value of the radius into our formula for area, we have

\begin{align*} A &= 4 \pi \cdot 14^2 \\[1em] &\approx 2463 \end{align*}

So the surface area of the sphere is $\hspace{0.2em} 2463 \text{ in}^2 \hspace{0.2em}$ .

Example

What is the radius of a sphere that has a surface area of $\hspace{0.2em} 36 \pi \hspace{0.2em}$ ?

Solution

Again, the surface area of a sphere is given by –

A = 4 \pi r^2

And the question tells us that the area is $\hspace{0.2em} 36 \pi \hspace{0.2em}$ . So,

Solving for $\hspace{0.2em} r \hspace{0.2em}$ , we get

\begin{align*} r &= \sqrt{\frac{36 \pi}{4 \pi}} \\[1.3em] &=3 \end{align*}

That's it.

And with that we come to the end of this tutorial. Until next time.