How to Find the Volume of a Pyramid

Only two things matter: how much floor the pyramid covers and how tall it stands. The shape of the base is irrelevant beyond its area, which is why one formula covers square pyramids, triangular ones and the great pyramids at Giza without modification.

Any base at all, as long as you can measure it

Most textbook pyramids sit on squares, so the base area is just one side squared. But nothing in the formula asks for a square. A pyramid on a 4 by 9 rectangle has a base area of 36, exactly like the 6 by 6 one, and at the same height it holds exactly the same 120. A triangular base works the same way, and so does a hexagonal one.

So the real work is often the base, not the pyramid. Get the base area by whatever route suits the shape, then the rest is one multiplication and one division. A cone is the same formula with a circle underneath it, which is why cones and pyramids of matching base area and height hold identical amounts.

Where the one third comes from

The division by three is the part that gets memorised and never understood, which is a shame because the demonstration is something you can hold in your hands. Build a box, then build a pyramid with the same base and the same height, and pour.

There is a cleaner version of the same fact for anyone who wants to see it without water. Take a cube of side 6, which holds 216. Three identical pyramids, each with a 6 by 6 base and a height of 6, assemble into that cube with no gaps. Each one holds 36 times 6 over 3, which is 72, and three lots of 72 is 216. The three pyramids meet at one corner of the cube and their apexes point in three different directions, which is why it is difficult to picture and completely convincing once built.

A worked check you can repeat on paper

Route three is worth a moment. A cone whose base circle has an area of 36 needs a radius of 3.385, and at a height of 10 it holds 120 as well. Pyramid and cone are the same formula wearing different bases, and the only reason the cone version looks different is that πr² has been substituted in for the base area.

The height is the vertical one

A pyramid has three different lengths that could reasonably be called its height, and only one of them belongs in the volume formula. The vertical height, 10 here, runs straight up from the centre of the base to the apex. The slant height, 10.44, runs up the middle of a triangular face. The sloping edge, 10.86, runs up a corner.

Which length	Value	Volume if you use it
Vertical height, the correct one	10.00	120.0
Slant height up a face	10.44	125.3
Sloping edge up a corner	10.86	130.4

The errors are only 4 and 9 percent, which is exactly what makes them dangerous. A wrong answer that is out by a factor of two announces itself. One that is out by 4 percent looks completely reasonable and sails through unchallenged. The slant heights belong to surface area calculations, where they are correct and necessary; they simply have no business in a volume.

Mistakes that catch people out

• Using a slant height. The volume wants the vertical height only. In the worked example the slant gives 125.3 instead of 120, which is wrong but not obviously wrong.
• Forgetting to divide by three. This gives 360, the volume of the box the pyramid sits inside. It is three times too big, and it is the most common slip by a distance.
• Using a base edge instead of a base area. The formula starts with an area in square units. Feeding in 6 rather than 36 gives 20, which is a sixth of the truth.
• Assuming the base must be square. Any base works. If it is a triangle, find its area with the triangle formula and carry on as normal.
• Answering in square units. Volume is cubic. If the units came out squared, a height went missing somewhere.

Try these

Common questions

What is the formula for the volume of a pyramid?

The base area times the vertical height, divided by three. A 6 by 6 base with a height of 10 gives 36 times 10 over 3, which is 120 cubic units.

Why do you divide by three?

Because a pyramid holds exactly one third of the box that shares its base and height. Three identical pyramids with a 6 by 6 base and a height of 6 assemble into a cube of side 6, and 3 times 72 is 216, the cube’s volume.

Does the base have to be a square?

No. Any flat shape works, and only its area enters the formula. A 4 by 9 base and a 6 by 6 base both have an area of 36, so at the same height they give the same volume of 120.

Which height do you use?

The vertical one, straight up from the centre of the base to the apex. Not the slant height up a face, and not the sloping edge up a corner. Using the slant here gives 125.3 instead of 120.

How do you find the height from the volume?

Multiply the volume by three, then divide by the base area. A volume of 120 on a base of 36 gives 360 over 36, so the height is 10.

Is a cone just a pyramid with a round base?

For volume purposes, yes. Both are a third of the base area times the height. The cone formula only looks different because πr² has been written in place of the base area.

Related lessons and calculators

• Surface area of a cone, the round cousin of this shape, where the slant height is the one you actually need.
• Area formulas for every common shape, for working out whatever base the pyramid happens to sit on.
• Area of a trapezoid, one of the more awkward bases to measure before the volume step.
• Radius of a cylinder, for the same rearranging habit applied to a solid with straight sides.
• Volume of 3D shapes calculator, which also handles a pyramid with its top cut off.
• the National Council of Teachers of Mathematics, for how solids and volume are sequenced in the curriculum.
• Volume of a sphere, the third member of the cylinder, sphere and cone ratio of 3 to 2 to 1.
• Volume of a cone, the round base version of this exact one third rule.
• Volume of a triangular prism, the shape that keeps the whole of base area times height instead of surrendering two thirds of it.
• Volume of a rectangular prism, the full box a pyramid of the same base and height fills only a third of.