Volume of Cylinders, Cones and Spheres

The three formulas

$$V_{\text{cylinder}} = \pi r^2 h \qquad V_{\text{cone}} = \tfrac{1}{3}\pi r^2 h \qquad V_{\text{sphere}} = \tfrac{4}{3}\pi r^3$$

Solid	Formula	Measurements needed
cylinder	$\pi r^2 h$	radius and height
cone	$\tfrac{1}{3}\pi r^2 h$	radius and height
sphere	$\tfrac{4}{3}\pi r^3$	radius only

A sphere has no height in its formula because a sphere has no second measurement — the radius describes it completely.

The cylinder is a stack of circles

$$V = \pi r^2 h = \underbrace{\pi r^2}_{\text{base area}} \times \underbrace{h}_{\text{height}}$$

That is the same idea as the volume of a rectangular prism: base area times height. The only change is that the base is a circle rather than a rectangle.

A cylinder of radius $3$ and height $10$:

$$V = \pi(3)^2(10) = 90\pi \approx 283 \text{ cubic units}$$

Why the cone is exactly a third

Take a cone and a cylinder with the same base and the same height. Fill the cone with water and pour it into the cylinder. It takes exactly three goes.

$$V_{\text{cone}} = \tfrac{1}{3} V_{\text{cylinder}} = \tfrac{1}{3}\pi r^2 h$$

The third is not an approximation and not a convention. It is exact, and it holds for every pair with matching base and height, however wide or tall.

The same relationship appears with a pyramid and its prism, for the same underlying reason: a solid that tapers uniformly to a point holds a third of the straight-sided solid around it.

The sphere and its cylinder

A sphere of radius $r$ fits snugly inside a cylinder of radius $r$ and height $2r$ — the narrowest and shortest cylinder that can contain it.

$$V_{\text{cylinder}} = \pi r^2 (2r) = 2\pi r^3$$ $$V_{\text{sphere}} = \tfrac{4}{3}\pi r^3$$

Divide one by the other:

$$\frac{\tfrac{4}{3}\pi r^3}{2\pi r^3} = \frac{4}{6} = \frac{2}{3}$$

A sphere is exactly two-thirds of the cylinder that encloses it. Archimedes found this and asked for it on his gravestone, which gives some sense of how much it pleased him.

It also makes the $\tfrac{4}{3}$ less arbitrary. It is whatever number makes the sphere two-thirds of $2\pi r^3$.

Why the powers differ

Quantity	Dimensions	Length appears
circumference	$1$	to the power $1$
area	$2$	to the power $2$
volume	$3$	to the power $3$

Volume comes from three lengths multiplied together, so every volume formula ends with lengths to the third power. In $\pi r^2 h$ those three are $r$, $r$ and $h$. In $\tfrac{4}{3}\pi r^3$ they are all $r$.

This decides what happens when a solid is scaled. Doubling the radius of a sphere multiplies its volume by $2^3 = 8$, not by $2$ — the same reason a dilation multiplies areas by $k^2$.

Working backwards

The formulas run in both directions.

A cylinder has volume $150\pi$ and height $6$. Find its radius.

$$\pi r^2 (6) = 150\pi$$

Divide both sides by $\pi$, then by $6$:

$$r^2 = 25 \quad\Rightarrow\quad r = 5$$

Leaving the answer in terms of $\pi$ until the end kept the arithmetic exact and made the $\pi$ cancel.

Worked examples

Common mistakes

Practice problems