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, $\hspace{0.2em} O \hspace{0.2em}$).

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.

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.

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.

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

$A = 4 \pi r^2$

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

$A_c = 4 \pi r^2$

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.

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