Volume of a Sphere

In this tutorial, we'll learn how to find the surface area of a sphere. And given how common spheres are in the world around us, all of us have some intuitive understanding of what a sphere is.

So, a sphere is a three-dimensional shape formed by the set of all points in space that are within a fixed distance (radius, usually denoted by $\hspace{0.2em} r \hspace{0.2em}$ ) from a certain fixed point (center, $\hspace{0.2em} O \hspace{0.2em}$ ).

Volume of a Sphere

As with any solid object, the volume of a sphere is a measure of the space it occupies.

Formula for Volume of a Sphere

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

V = \frac{4}{3} \pi r^3

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Derivation of the Formula

The great Greek polymath, Archimedes, proved that the volume of a sphere is two-thirds the volume of a cylinder with the same radius and height.^#

So,

V_{sphere} = \frac{2}{3}V_{cylinder} \hspace{0.25cm} \rule[0.1cm]{1cm}{0.1em} \hspace{0.15cm} (1)

So let’s consider a sphere of radius $\hspace{0.2em} r \hspace{0.2em}$ . Its height would be the same as its diameter – $\hspace{0.2em} 2r \hspace{0.2em}$ .

Now, the formula for the volume of a cylinder with a radius $\hspace{0.2em} r \hspace{0.2em}$ and height $\hspace{0.2em} h \hspace{0.2em}$ is –

V_{cylinder} = \pi r^2 h

Substituting $\hspace{0.2em} h= 2r \hspace{0.2em}$ gives us

\begin{align*} V_{cylinder} &= \pi \cdot r^2 \cdot 2r \\[1em] V_{cylinder} &= 2 \pi r^3 \hspace{0.25cm} \rule[0.1cm]{1cm}{0.1em} \hspace{0.15cm} (2) \end{align*}

From $\hspace{0.2em} (1) \hspace{0.2em}$ and $\hspace{0.2em} (2) \hspace{0.2em}$ , we get the volume of the sphere.

\begin{align*} V_{sphere} &= \frac{2}{3} \cdot 2 \pi r^3 \\[1.3em] &= \frac{4}{3} \pi r^3 \end{align*}

How to Find the Volume of a Sphere – Examples

So far so good. Now let's use what we have learned so far and solve a couple of examples.

Example

Find the volume of a sphere with a radius of $\hspace{0.2em} 6 \hspace{0.2em}$ cm.

Solution

The question tells us the radius is $\hspace{0.2em} 6 \hspace{0.2em}$ cm. Substituting this value in the formula for the volume of a sphere, we get -

\begin{align*} V &= \frac{4}{3} \pi r^3 \\[1.3em] &= \frac{4}{3} \pi \cdot 6^3 \\[1.3em] &\approx 904.78 \end{align*}

So the volume of the sphere is $\hspace{0.2em} 904.78 \hspace{0.2em}$ cubic centimeters $\hspace{0.2em} (\text{cm}^3) \hspace{0.2em}$ .

Example

The volume of a sphere is $\hspace{0.2em} 36π \hspace{0.2em}$ . Find its diameter.

Solution

We have,

V = \frac{4}{3} \pi r^3

Substituting the value of $\hspace{0.2em} V \hspace{0.2em}$ in the formula and solving for $\hspace{0.2em} r \hspace{0.2em}$ , we get

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

But the question asks us to find the diameter, so there's one last step.

\begin{align*} d &= 2r \\[1em] &= 6 \end{align*}

And that brings us to the end of this tutorial on the volume of a sphere. Until next time.

Footnotes

^# In fact, Archimedes proved that the sum of the volumes of a cone and a sphere was equal to the volume of a cylinder – each on the same base and with the same height. And that their volumes were in the ratio of $\hspace{0.2em} 1:2:3$ .

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