Volume of Prisms & Cylinders

Flat tops: prisms and cylinders use $V = Bh$

A prism or a cylinder is a base shape stacked straight up to some height, so its volume is base area times height: $V = Bh$. For a rectangular prism the base is a rectangle, giving $V = \ell w h$. For a triangular prism the base is a triangle, so find that triangle's area first, then multiply by the prism's length.

A cylinder is the same idea with a circular base. The base area is $\pi r^2$, so $V = \pi r^2 h$ — as shown below, $r$ is the radius of the circular base and $h$ is the height between the two bases.

Pointed tops: cones and pyramids take one third

A cone is a cylinder that tapers to a point, and it holds exactly one third of the matching cylinder's volume: $V = \dfrac{1}{3}\pi r^2 h$. The same relationship holds for pyramids and prisms: a pyramid holds one third of the prism with the same base and height, so $V = \dfrac{1}{3}Bh$.

One warning for cones and pyramids: the $h$ in the formula is the vertical height, from the apex straight down to the center of the base — never the slant height along the side. If a problem gives the slant height, use the Pythagorean theorem with the radius to find the vertical height first.

The sphere

A sphere has no base and no height, so it gets its own formula: $V = \dfrac{4}{3}\pi r^3$. Notice the radius is cubed, not squared — volume is three-dimensional, so a sphere's volume grows with the cube of its radius. Doubling the radius makes the volume $8$ times bigger.

For a hemisphere — half a ball — take half: $V = \dfrac{2}{3}\pi r^3$.

Worked examples

Try one yourself

Common questions