Pyramid volume:
formula and worked example

What the volume of a pyramid tells you

A pyramid with a square base takes up exactly one third of the space of a box that has the same base and the same height. That single fact is why the formula is so short, and it is also the quickest sanity check you can run on any answer: the pyramid volume must always be a third of base area times height, never more.

People reach for this number when they need to know how much material a pointed shape holds or displaces. Typical cases are a heap of sand or gravel that has settled into a pyramid, a hip-roof loft space, a monument or planter model, a packaging mock-up, or a classroom problem about the Egyptian pyramids. In each case the volume is the first step; multiply by density to get mass or by a rate per cubic metre to get cost.

Two measurements are enough. You need the length of one side of the square base and the vertical height from the tip down to the middle of the base.

The formula and what each input means

The rule is volume equals base side squared, times height, divided by three. Squaring the side gives the base area, which is the square footprint of the pyramid.

• Base side: the length of one edge of the square base, in any length unit such as metres or centimetres.
• Height: the perpendicular distance from the apex to the base plane, in the same unit as the base side.
• Result: a volume in that unit cubed, for example cubic metres.

Mixing units is the easiest way to get a wrong answer here, because the height is multiplied in only once while the side is squared. Convert both to one unit before you start.

Worked example: a 6 by 9 pyramid

Take a pyramid whose square base measures 6 on each side and whose vertical height is 9.

1. 1Square the base side: 6 × 6 = 36, which is the base area.
2. 2Multiply by the height: 36 × 9 = 324, the volume of the full box.
3. 3Divide by three: 324 ÷ 3 = 108 cubic units.

As a quick check, the box that encloses this pyramid holds 324, and 108 is exactly a third of that, so the arithmetic is consistent.

Reading the number: from cubic units to something usable

The raw answer is only as meaningful as the unit behind it. If the 6 and 9 were metres, the pyramid holds 108 cubic metres, which is 108,000 litres of space. If they were centimetres, it is 108 cubic centimetres, which is about 108 millilitres. Always attach the unit before you quote the figure.

For a physical object, the next step is usually to convert to mass or cost. Multiply the volume by the material's bulk density to get weight, and keep in mind that loose heaps of soil or grain settle into angles that are not always a perfect pyramid, so a measured volume can differ from the ideal shape by a margin that depends on the material.

For a roof void or a tent, the volume tells you how much air has to be heated or ventilated. Here the walls and floor are not part of the answer, so the figure is the interior volume only if you measured the inside dimensions.

What happens when the base side or the height changes

Because the base side is squared, the volume is much more sensitive to it than to the height. Make the base 10% wider and the footprint grows by 21%, so the volume does too. Make the height 10% taller and the volume rises by exactly 10%.

This is worth remembering when you are measuring on site. A small error in the side length matters twice as much as the same error in height, so spend your care on the base.

Slips that quietly ruin a pyramid volume

• Using the slant height instead of the vertical height. The slant runs along a face and is always longer than the vertical height, so the volume comes out too large.
• Forgetting to divide by three, which gives the volume of a prism, not a pyramid.
• Treating a rectangular base as a square. This rule assumes equal sides; with a length and a width, use their product in place of the square.
• Mixing centimetres and metres in the same calculation, then reading the answer in the wrong cubic unit.

A short checklist before you trust the figure

Suppose you are ordering sand for a pyramid-shaped display and the supplier quotes per cubic metre. Run through these steps first.

1. 1Measure the base side and the vertical height in metres.
2. 2Compute base side squared times height, then divide by three.
3. 3Compare with the enclosing box: your answer must be exactly one third of it.
4. 4Add a modest allowance for settling or waste, agreed with the supplier.