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Volume of Pyramids & Cones

A pyramid or cone holds exactly one-third of the prism or cylinder with the same base and height. That one fact — the factor of 13\dfrac{1}{3} — is the whole lesson.

The formula is V=13BhV = \dfrac{1}{3} B h, where BB is the area of the base and hh is the vertical height. Find the base area the usual way, multiply by the height, and take a third.

One formula, two shapes

For any pyramid, BB is the area of its polygon base; for a cone, B=πr2B = \pi r^2 because the base is a circle. In both cases V=13BhV = \dfrac{1}{3} B h.

So a cone is V=13πr2hV = \dfrac{1}{3}\pi r^2 h and a square pyramid is V=13s2hV = \dfrac{1}{3} s^2 h. Same idea, different base area plugged in.

Use vertical height, not slant height

Volume always uses the straight-up height hh from base to apex — never the slant height. If a problem gives slant height, find hh first with the right triangle inside the solid.

This is the opposite of surface area, which needs slant height. Keep them straight: volume wants vertical, surface area wants slant.

The dashed segment in the cone below is the vertical height hh — the straight-up distance the volume formula needs, not the slanted distance along the outside.

hh
rr

Worked examples

Example 1: a square pyramid

A square pyramid has base edge 66 and height 1010. Find its volume.

Volume is one-third base times heightV=13s2hV = \tfrac{1}{3} s^2 h
Substitute13(6)2(10)\tfrac{1}{3}(6)^2(10)
Simplify13(360)=120\tfrac{1}{3}(360) = 120

Answer: 120120 cubic units

Example 2: a cone

Find the volume of a cone with radius 33 and height 77.

One-third circle base times heightV=13πr2hV = \tfrac{1}{3}\pi r^2 h
Substitute13π(3)2(7)\tfrac{1}{3}\pi(3)^2(7)
Simplify13π(63)=21π\tfrac{1}{3}\pi(63) = 21\pi

Answer: 21π21\pi cubic units

Example 3: a cone given the slant height

Find the volume of a cone with radius 33 and slant height 55.

Volume needs vertical height, so find it firsth=ℓ2−r2h = \sqrt{\ell^2 - r^2}
Substitute the slant height and radiush=52−32=16=4h = \sqrt{5^2 - 3^2} = \sqrt{16} = 4
One-third circle base times height13π(3)2(4)\tfrac{1}{3}\pi(3)^2(4)
Simplify13π(36)=12π\tfrac{1}{3}\pi(36) = 12\pi

Answer: 12π12\pi cubic units

Try one yourself

h=10h=10
66

Common questions

Where does the one-third come from?

It takes exactly three pyramids (or cones) to fill the matching prism (or cylinder) with the same base and height. That is a fact you can demonstrate by pouring water, and it gives the 13\tfrac{1}{3}.

Slant height or vertical height for volume?

Vertical height, always. Volume measures how tall the solid actually stands, which is the straight-up distance from base to apex.

What are the units?

Cubic units — cubic inches, cubic centimeters — because volume fills space in three dimensions.

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