A pyramid is one of those shapes everyone recognises but few people can measure. The Great Pyramid of Giza, a tent, a pointed roof, a pile of sand — all hold a definite amount of space, found in two steps: work out the area of the base, then multiply by the height and take a third. The Volume Calculator does it in seconds, but the hand method is easy to learn.
What is a pyramid?
A pyramid is a solid with a flat base and triangular faces that meet at a single point, called the apex. The base can be any polygon: a square gives a square pyramid, a rectangle gives a rectangular pyramid, a triangle gives a triangular pyramid. Square and rectangular bases are the common ones — roofs, tents, monuments.
Whichever base it has, the method is the same: the formula only cares about base size and height. The height is the straight vertical distance from the base to the apex — not a sloping edge, and not the height of a triangular face.
The pyramid volume formula
The formula for the volume of any pyramid is:
V = ⅓ × base area × height
In symbols, V = ⅓Bh. It works for every pyramid, because the base area already carries the base shape.
The method has three steps. First, find the area of the base: for a square base with side s, that is s²; for a rectangular base with length l and width w, it is l × w; for a triangular base with base b and triangle height hₜ, it is ½ × b × hₜ. Second, multiply the base area by the pyramid’s vertical height. Third, divide by 3.
Units follow the measurements: centimetres in, cubic centimetres (cm³) out. Every measurement must use the same unit first — a base in centimetres and a height in metres gives a wrong answer.
Why is there a ⅓ in the formula?
The ⅓ is not a fudge factor; it is the exact share of the box a pyramid fills. A square pyramid with base 4 × 4 and height 6 sits inside a box of 4 × 4 × 6 = 96 cubic units, and holds 32 — exactly one third. The same ratio holds for any pyramid and its enclosing prism.
Three identical pyramids fit together to make that box, so each is a third of it. The cone follows the same rule: radius 3, height 6 → 56.55 cubic units, while the cylinder around it holds 169.65 — and 169.65 ÷ 3 = 56.55.
That is also a handy check: work out the box with the same base and height — if your pyramid answer equals the box volume, you forgot the ⅓, and if it is bigger, something else is wrong.
Square-base pyramids
The square pyramid is the most common case: V = ⅓s²h. Square the side first — that is the base area — then multiply by the height, then divide by 3.
Worked example — square base 4 × 4, height 6. Base area 4² = 16. Times height: 16 × 6 = 96. A third: 96 ÷ 3 = 32 cubic units. The box holds 96, and 32 is a third of it.
Worked example — square base 5 × 5, height 7. Base area 5² = 25. Times height: 25 × 7 = 175. A third: 175 ÷ 3 = 58.33 cubic units, rounded to two decimal places. Round only at the end.
Rectangular-base pyramids
A rectangular pyramid works exactly like a square one, except the base area is length × width. The formula is V = ⅓ × l × w × h: multiply the three numbers together, then divide by 3.
Worked example — rectangular base 6 × 3, height 9. Base area 6 × 3 = 18. Times height: 18 × 9 = 162. A third: 162 ÷ 3 = 54 cubic units — one third of the 162 held by the enclosing box.
Triangular-base pyramids
A triangular pyramid has a triangle for its base, so first find its area: ½ × base × height of the triangle. The formula is V = ⅓ × (½ × b × hₜ) × h, where hₜ is the height of the triangle and h is the vertical height of the pyramid — two different measurements that must not be confused.
Worked example — triangular base with base 6 and triangle height 4, pyramid height 9. Base area ½ × 6 × 4 = 12. Times pyramid height: 12 × 9 = 108. A third: 108 ÷ 3 = 36 cubic units.
More worked examples
A pyramid roof. A square pyramid roof has a 6 × 6 metre base and a vertical height of 4 metres. Base area 6² = 36. Times height: 36 × 4 = 144. A third: 144 ÷ 3 = 48 cubic metres of air under the roof.
In centimetres and litres. A pyramid-shaped container has a 40 × 40 centimetre base and a height of 60 centimetres. Base area 40² = 1,600. Times height: 1,600 × 60 = 96,000. A third: 96,000 ÷ 3 = 32,000 cubic centimetres — and since 1,000 cm³ make 1 litre, the container holds 32 litres.
Working backwards. Sometimes the volume is known and the height is not. A square pyramid has a base 4 × 4 and a volume of 32 cubic units. Rearrange the formula: height = 3 × volume ÷ base area = 3 × 32 ÷ 16 = 6.
Slant height vs vertical height
The slant height is the height of one triangular face, from the midpoint of a base edge to the apex. The vertical height runs straight down from the apex to the centre of the base. With half the base side they form a right-angled triangle: vertical height² + (half the base side)² = slant height².
Worked example — finding the vertical height from the slant height. A square pyramid has a base side of 6 and a slant height of 5. Half the base side is 3, so the vertical height is √(5² − 3²) = √(25 − 9) = √16 = 4. Volume: base area 6² = 36, times 4 = 144, divided by 3 = 48 cubic units.
Using the slant height of 5 as the vertical height would give ⅓ × 36 × 5 = 60 — a fifth too big. Convert a slant height first; never put it straight into the volume formula.
Common mistakes
- Forgetting the ⅓. Multiplying base area by height and stopping gives three times too much: 96 instead of 32. Always divide by 3.
- Using the slant height instead of the vertical height. The slant height is always longer, so the answer comes out too big — 60 instead of 48 above. Convert it first with Pythagoras.
- Using the base length instead of the base area. The formula needs the area of the base, not one side. For base side 4, the area is 16, not 4 — putting 4 in gives ⅓ × 4 × 6 = 8.
- Mixing up the two heights in a triangular pyramid. A triangular pyramid has a triangle height (for the base area) and a pyramid height (for the volume). Label them: hₜ for the triangle, h for the pyramid.
- Mixing units. A base in centimetres and a height in metres cannot go into the same formula. Convert first — 1 metre is 100 centimetres.
- Confusing volume with surface area. Volume is the space inside, in cubic units; surface area is the skin outside, in square units. Neither substitutes for the other.
Pyramid volume and the other shapes
A cone is a pyramid with a circular base: its volume is ⅓πr²h, and a cone with radius 3 and height 6 holds 56.55 cubic units — one third of the 169.65 held by the cylinder with the same radius and height.
For comparison, a cube with side 4 holds 64, a sphere with radius 2 holds 33.51, and a cylinder with radius 3 and height 5 holds 141.37.
Related tools
For the exact answer on any measurements, the Volume Calculator works out the volume of a square pyramid — and of a cube, rectangular prism, cylinder, cone and sphere — showing the formula with your numbers substituted in. It is free and needs no sign-in. For all of ToolZana’s free tools, start at the ToolZana home page.
The short version
The pyramid volume formula is V = ⅓ × base area × height: square base s², rectangular base l × w, triangular base ½ × b × hₜ — one third of the prism with the same base and height. Square base 4 × 4, height 6 → 32 cubic units; rectangular base 6 × 3, height 9 → 54; triangular base 6 × 4, height 9 → 36. Always use the vertical height, never the slant height, and always divide by 3. For any other measurements, the Volume Calculator gives the exact answer in seconds.
Frequently Asked Questions
What is the pyramid volume formula?
The pyramid volume formula is V = ⅓ × base area × height, where the height is the vertical distance from base to apex. For a square pyramid it becomes V = ⅓s²h; for a rectangular pyramid, V = ⅓ × l × w × h; for a triangular pyramid, V = ⅓ × (½ × b × hₜ) × h.
Why is there a ⅓ in the pyramid volume formula?
Because a pyramid fills exactly one third of the prism with the same base and height — three identical pyramids make the box that encloses one of them. A square pyramid with base 4 × 4 and height 6 holds 32 cubic units, while the box around it holds 4 × 4 × 6 = 96.
How do I find the volume of a pyramid with a rectangular base?
Work out the base area as length × width, multiply by the vertical height, and divide by 3. For a base of 6 × 3 and a height of 9, the base area is 6 × 3 = 18, times the height is 18 × 9 = 162, and a third of that is 162 ÷ 3 = 54 cubic units.
How do I find the volume of a triangular pyramid?
First find the area of the triangular base: ½ × base × height of the triangle. Then multiply by the pyramid’s vertical height and divide by 3. For a triangle base 6, triangle height 4 and pyramid height 9, the base area is ½ × 6 × 4 = 12, times 9 is 108, and a third is 36 cubic units.
What is the difference between slant height and vertical height?
The vertical height is the straight distance from the base to the apex — the height the volume formula needs. The slant height is the height of one triangular face, from the midpoint of a base edge to the apex, and it is always longer. They are linked by Pythagoras: vertical height² + (half the base side)² = slant height².
How do I work out a pyramid’s height if I know its volume?
Rearrange the formula: height = 3 × volume ÷ base area. A square pyramid with a base of 4 × 4 and a volume of 32 has height 3 × 32 ÷ 16 = 96 ÷ 16 = 6. The same rearrangement finds any missing measurement: base area = 3 × volume ÷ height.
What units is pyramid volume measured in?
Pyramid volume is measured in cubic units — cm³, m³, in³ or ft³ — matching the unit of the measurements. A base of 40 × 40 cm and a height of 60 cm gives 32,000 cm³, which is 32 litres, since 1,000 cm³ make 1 litre.
Can I use the pyramid volume formula for a cone?
Yes, with one change: a cone is a pyramid with a circular base, so its base area is πr² and its volume is V = ⅓πr²h. A cone with radius 3 and height 6 holds ⅓ × π × 9 × 6 = 56.55 cubic units — one third of the 169.65 held by the cylinder with the same radius and height.
How do I find the volume of a pyramid-shaped roof?
Treat the roof as a square or rectangular pyramid: measure the base of the roof space and its vertical height, then apply V = ⅓ × base area × height. A square pyramid roof with a 6 × 6 metre base and a height of 4 metres holds ⅓ × 36 × 4 = 48 cubic metres of air.
How can I check my pyramid volume answer quickly?
Work out the volume of the box with the same base and height: your pyramid answer should be exactly one third of it. For base 4 × 4 and height 6, the box holds 96, so the pyramid must hold 32 — if your answer is 96, you forgot the ⅓. For a fast check, the free Volume Calculator shows the formula with your numbers substituted in.