Volume of Cones & Spheres

Cones and spheres each get their own volume formula. A cone is an ice-cream-cone shape — a circular base that narrows to a point. A sphere is a ball — perfectly round in every direction. Their formulas are $\displaystyle V = \frac{1}{3} \pi r^2 h$ for a cone and $\displaystyle V = \frac{4}{3} \pi r^3$ for a sphere.

Both formulas connect back to the cylinder. A cone with the same base and height as a cylinder holds exactly one third of the cylinder's volume — pour three cones of water into the matching cylinder and it fills to the top. The sphere has no separate height at all; its size is set entirely by the radius, which is why $r$ shows up cubed.

The two formulas

Cone: $\displaystyle V = \frac{1}{3} \pi r^2 h$ . It's the cylinder formula $\pi r^2 h$ with a $\displaystyle \frac{1}{3}$ in front, because the cone narrows to a point instead of keeping its full base all the way up. The cone below is labeled with the two measurements it needs — the base radius $r$ and the height $h$ .

Sphere: $\displaystyle V = \frac{4}{3} \pi r^3$ . There is no $h$ in this formula — a sphere is the same distance from its center in every direction, so the radius does all the work.

Keep the fractions straight: the cone takes $\displaystyle \frac{1}{3}$ , the sphere takes $\displaystyle \frac{4}{3}$ . Writing the formula down before substituting is the cheapest insurance there is.

Squared vs. cubed

In the cone formula the radius is squared, because $\pi r^2$ is the area of the circular base. In the sphere formula the radius is cubed. If you catch yourself writing $r^2$ inside a sphere problem, stop and fix it — $3^3 = 27$ , not $9$ , and that difference changes everything downstream.

A sphere has just one measurement, the radius $r$ shown below — no separate height — which is why $r$ is the only length in its formula.

As with cylinders, watch for the diameter trap: if the problem gives a diameter, halve it to get the radius before you substitute.

Worked examples

Example 1: volume of a cone

Find the volume of a cone with radius $2$ and height $6$ . Use $\pi \approx 3.14$ .

Write the formula

V = \dfrac{1}{3} \pi r^2 h

Substitute the values

V = \dfrac{1}{3} \cdot 3.14 \cdot 2^2 \cdot 6

Square the radius

V = \dfrac{1}{3} \cdot 3.14 \cdot 4 \cdot 6

Multiply, taking one third at the end

V = \dfrac{1}{3} \cdot 75.36 = 25.12

Answer: $V = 25.12$

Example 2: volume of a sphere

Find the volume of a sphere with radius $3$ . Use $\pi \approx 3.14$ .

Write the formula

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

Substitute the values

V = \dfrac{4}{3} \cdot 3.14 \cdot 3^3

Cube the radius — not square it

V = \dfrac{4}{3} \cdot 3.14 \cdot 27

Multiply

V = 113.04

Answer: $V = 113.04$

Example 3: a cone with the diameter given

Find the volume of a cone with diameter $6$ and height $10$ . Use $\pi \approx 3.14$ .

Halve the diameter to get the radius

r = \dfrac{6}{2} = 3

Write the formula

V = \dfrac{1}{3} \pi r^2 h

Substitute the values

V = \dfrac{1}{3} \cdot 3.14 \cdot 3^2 \cdot 10

Square the radius, then multiply

V = \dfrac{1}{3} \cdot 282.6 = 94.2

Answer: $V = 94.2$

Try one yourself

Common questions

Why does the cone formula have a 1/3 in it?

A cone with the same base and height as a cylinder holds exactly one third of the cylinder's volume. The cone narrows to a point, so most of the matching cylinder is empty space around it — one third full, two thirds empty.

Why is the radius cubed for a sphere but squared for a cone?

The cone's $r^2$ comes from the area of its circular base, and the height supplies the third dimension. A sphere has no separate height — the radius controls all three dimensions — so it appears three times: $r^3$ .

What if the problem gives the diameter?

Divide by $2$ first. Both formulas need the radius. A ball with a diameter of $6$ inches has $r = 3$ , and $3$ is what gets cubed.

How do I keep 1/3 and 4/3 straight?

Tie each fraction to its shape: the cone is the smaller, pointier solid, so it gets the smaller fraction $\displaystyle \frac{1}{3}$ . The sphere bulges out in every direction and gets $\displaystyle \frac{4}{3}$ . Writing the formula before substituting catches the mix-up early.

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Volume of Cones & Spheres

The two formulas

Squared vs. cubed

Worked examples

Example 1: volume of a cone

Example 2: volume of a sphere

Example 3: a cone with the diameter given

Try one yourself

Common questions

Keep learning

Volume of Cylinders

Surface Area of Prisms & Cylinders

Missing Dimensions & Composite Solids