Volume of a Prism

Volume measures the space a solid fills, and the honest way to picture it is filling the solid with little cubes. Each cube is one centimetre in every direction, so each one is one cubic centimetre. This box is 4 cubes across, 3 cubes back and 2 cubes high. Build it a layer at a time: the bottom layer is 4 across by 3 back, which is 12 cubes, and there are 2 layers, so 24 cubes in total. The volume is 24 cubic centimetres. Notice the unit — you multiplied three lengths, so it is cubed, not squared. And you never actually count the cubes one by one. Twelve in a layer times two layers does the job.

For a rectangular prism — a box — the formula is length times width times height. This one is 6 centimetres wide, 4 deep and 5 high, so its volume is 6 times 4 times 5, which is 120 cubic centimetres. You can multiply the three sides in any order. There is a second way to see it that is worth more later. Work out the area of the bottom face first: 6 times 4 is 24 square centimetres. Then stack that layer 5 high: 24 times 5 is 120 again. Find the area of one face, then multiply by how far the solid runs. A cube is just a box with equal edges, so a cube of side 3 holds 27 cubic centimetres.

Here is the idea that ties the whole topic together. Because a prism has the same cross-section all the way through, you can think of it as a stack of identical thin slices. Each slice covers the cross-section area, so if you know that area and you know how far the prism runs, the volume is simply the area of the cross-section multiplied by the length. Written short, V equals A times L. If a prism has a cross-section of 18 square centimetres and a length of 7 centimetres, its volume is 18 times 7, which is 126 cubic centimetres — and notice you never needed to know what shape that slice was. This one rule handles boxes and triangular prisms alike.

A triangular prism uses the same rule, and the only new work is finding the area of a triangle — which you already know is a half times base times height. Take a prism whose triangle has a base of 8 centimetres and a perpendicular height of 5, and which runs 9 centimetres back. The cross-section area is a half times 8 times 5, which is 20 square centimetres, and the volume is 20 times 9, so 180 cubic centimetres. Two warnings. The prism has two measurements you might call a height, so label the triangle's h and the prism's length L separately. And do not lose the half, because without it you get 360 — the volume of the whole box the prism sits inside.

One last trap. A triangular prism does not have to stand with the triangle facing you. This one is lying like a tent, resting on a rectangular face, and it is tempting to treat that rectangle as the cross-section. It is not. The slice that repeats is the triangle at each end, so the working is exactly the same as before: a half times 6 times 4 gives 12 square centimetres, then 12 times 10 gives 120 cubic centimetres. To find the cross-section of any prism, look for the face that appears twice, once at each end, with straight edges joining the two. That pair is your slice. Then area of the slice, times the length, and cubed units on the answer.