How it works
V = ⅓ × base area × perpendicular height. Polygon base area = n × side² ÷ (4 × tan(π/n)). Square frustum: V = h ÷ 3 × (A1 + A2 + √(A1 × A2)).
- Choose the base shape and one length unit.
- Enter the base measurements and perpendicular height.
- For square pyramids, you can use the face slant height instead.
- Read the volume, base area and working.
Worked example: A square base with side 6 cm has area 36 cm². With perpendicular height 9 cm, V = ⅓ × 36 × 9 = 108 cm³, or 0.108 L.
Example pyramid volumes
| Base shape | Pyramid height | Volume | Base area | Base measurements |
|---|---|---|---|---|
| Square | 9 cm | 108 cm³ | 36 cm² | Side 6 cm |
| Rectangle | 6 cm | 40 cm³ | 20 cm² | 4 × 5 cm |
| Triangle | 10 cm | 40 cm³ | 12 cm² | Base 6, triangle height 4 cm |
| Regular polygon | 10 cm | 34.641 cm³ | 10.3923 cm² | 6 sides, side 2 cm |
| Frustum (square) | 3 cm | 28 cm³ | 16 cm² | Bottom 4, top 2 cm |
Swipe the table sideways to see the rest.
Things to consider
- Use perpendicular height for volume.
- Do not confuse a face slant with a sloping edge.
- A regular polygon has equal sides and angles.
- Triangle base height is not pyramid height.
- Frustum top and bottom faces must be parallel.
Frequently asked questions
What is the pyramid volume formula?
V = ⅓ × base area × perpendicular height. Square bases use side², rectangles use length × width, and triangles use base × triangle height ÷ 2. Use one length unit throughout.
Why is a pyramid one third of a prism?
At a fraction t of the height below the apex, a cross-section has area t² times the base area. Adding these slices through the height gives base area × height × the integral of t² from 0 to 1, which is one third. A prism keeps the same area through its full height.
What is the difference between slant height and perpendicular height?
Perpendicular height runs from the apex at a right angle to the base plane. A square pyramid's face slant height runs to the midpoint of a base edge. For a right square pyramid, h = √(slant² − (side/2)²). The slant must be greater than half the side. This is not the apex-to-corner edge length.
How do I calculate triangular pyramid or tetrahedron volume?
A triangular pyramid has four triangular faces and is a tetrahedron. Its volume is ⅓ × triangle base area × perpendicular pyramid height, or base × triangle height × pyramid height ÷ 6. The two heights are different measurements.
What was the approximate original volume of the Great Pyramid of Giza?
Using an approximate square base side of 230.3 m and original perpendicular height of 146.6 m gives 230.3² × 146.6 ÷ 3, about 2.59 million m³. This is a geometric estimate of the original exterior shape, not a precise survey or the amount of solid stone.
How do I calculate a square pyramid frustum?
Use V = h ÷ 3 × (A1 + A2 + √(A1 × A2)), where A1 and A2 are the bottom and top square areas. With bottom side 4, top side 2 and height 3, V = 28 cubic units. The top cannot be larger than the bottom in this calculator.
How do I compare a pyramid with a cone or box?
A cone also has one third of base area times height, but its base is circular. Use the Cone Volume Calculator for cones or the Rectangular Prism Volume Calculator for boxes.
Results are estimates for general information only, not financial, medical or professional advice.