Quick answer: the volume of any pyramid is V = (1/3) × base area × height. Compute the base area for your base shape (square: a², rectangle: l×w, triangle: ½bh), multiply by the vertical height, divide by three. Know the base dimensions instead of the area? Use the dedicated square, rectangular or triangular pyramid calculators.
One formula, every pyramid
where A = area of the base · h = vertical height (base plane to apex)
The remarkable thing about this formula is what it ignores: the base's shape. Square, star-shaped, or a lopsided pentagon — if the solid tapers linearly from that base to a single point, its volume is one-third of base area × height. Even the apex position doesn't matter: an oblique pyramid whose tip leans off-center holds exactly as much as an upright one of the same height (Cavalieri's principle).
Base-area formulas for the common cases:
| Base shape | Base area A | Full volume formula |
|---|---|---|
| Square (edge a) | a² | V = (1/3) a² h |
| Rectangle (l × w) | l w | V = (1/3) l w h |
| Triangle (base b, height h_b) | ½ b h_b | V = (1/6) b h_b h |
| Regular pentagon (side a) | 1.720 a² | V = 0.573 a² h |
| Regular hexagon (side a) | 2.598 a² | V = 0.866 a² h |
| Regular octagon (side a) | 4.828 a² | V = 1.609 a² h |
Worked example: the Great Pyramid of Giza
The most famous pyramid on Earth, as originally built:
- Base: a square 756 ft on each side → A = 756² = 571,536 ft²
- Height: 481 ft
- V = (1/3) × 571,536 × 481 ≈ 91.6 million ft³ ≈ 2.6 million m³
That's roughly 1,040 Olympic swimming pools of limestone, placed one block at a time. Enter A = 571536 ft² and h = 481 ft above to see the calculator reproduce it.
Where pyramid volume shows up in real life
Hip roofs and skylights
A pyramid-hipped roof over a 30 × 30 ft square, rising 8 ft to the peak, encloses (1/3) × 900 × 8 = 2,400 ft³ of attic — a number that drives insulation and ventilation sizing.
Hoppers and bins
Rectangular bins drain through inverted pyramids. A hopper narrowing from a 4 × 6 ft opening over 3 ft holds (1/3) × 24 × 3 = 24 ft³ in the tapered section. If it narrows to a rectangular chute rather than a point, calculate the full pyramid and subtract the missing tip.
Landscape features and monuments
A glass pyramid entrance like the Louvre's (35.4 m square base, 21.6 m tall) encloses (1/3) × 1,253 × 21.6 ≈ 9,000 m³ of air-conditioned space.
Common mistakes & pro tips
- Slant height used as height. The height in the formula is vertical. For a square pyramid, h = √(s² − (a/2)²) where s is the face slant height.
- Perimeter confused with area. The formula needs base area in square units. A 10-ft-square base has A = 100 ft², not 40.
- Forgetting the ÷3. Base × height alone gives the enclosing prism — three times too much.
- Pro tip — truncated pyramids. A pyramid with the top sliced off (a frustum) is the full pyramid minus the small pyramid removed. Compute both with this calculator and subtract.
- Pro tip — mixed units. Enter the base area in ft² and the height in inches if that's how you measured; the per-field unit selectors reconcile them.
How to use this calculator
- Find the base area. Square base: side². Rectangular: length × width. Triangular: ½ × base × height. Any polygon works — the volume formula doesn't care about the base's shape, only its area.
- Measure the vertical height. Use the perpendicular distance from the base plane to the apex — not the slanted edge or face.
- Let it apply V = (1/3) A h. Multiply base area by height and divide by three. The calculator substitutes your numbers live.
- Convert if needed. Switch the result between cubic units, liters and gallons from the output menu.
Frequently asked questions
What is the formula for the volume of a pyramid?
V = (1/3) × base area × height, for any pyramid regardless of base shape. A square-based pyramid becomes V = (1/3)a²h; rectangular becomes V = (1/3)lwh; triangular becomes V = (1/6) × base × triangle-height × pyramid-height.
Why is pyramid volume one-third of a prism?
Because the cross-section shrinks with the square of the distance from the apex, and those squares integrate to exactly one-third of the full base. It's the same rule that makes a cone one-third of its cylinder — a cone is just a pyramid with a round base.
What is the volume of the Great Pyramid of Giza?
Originally about 756 ft on each side and 481 ft tall: V = (1/3) × 756² × 481 ≈ 91.6 million cubic feet, or roughly 2.6 million cubic meters — enough stone to fill about 1,040 Olympic swimming pools.
Do I use the slant height or the vertical height?
Always the vertical height. For a square pyramid, convert slant height s (measured up the middle of a face) with h = √(s² − (a/2)²), where a is the base edge.
How do I find the volume of a hexagonal pyramid?
Compute the hexagon's area first — for a regular hexagon with side a, area = 2.598a² — then multiply by the pyramid height and divide by 3. This calculator accepts the base area directly, so any polygon base works.