Volume of a Square Pyramid Calculator
Enter the square base edge and either the vertical height or the slant height (pick the mode) to get volume, base area, and both height measurements in the same units you entered.
Example: with Base edge (a) 6 · Which height do you know? Vertical height (apex straight up from center) · Height value 8 → Volume: 96 cubic units.
- Base area (a²)36 square units
- Vertical height h8 units
- Slant height ℓ8.544 units
Computed by the calculator below using its default values. Change any input to see your own numbers.
V = a²h/3 — a pyramid fills exactly one-third of the box around it.
The one-third rule
A pyramid holds exactly one-third of the prism that shares its base and height: V = a²h/3 for a square base. The factor is not an approximation — three identical oblique pyramids assemble perfectly into a cube, and calculus confirms the same 1/3 for every base shape. That is why the cone formula πr²h/3 looks so similar.
For the default 6 × 6 base and height 8, the base area is 36, the surrounding box would hold 288, and the pyramid holds 96 cubic units.
Vertical height versus slant height
The vertical height h drops from the apex straight down to the center of the base. The slant height ℓ runs from the apex down the middle of a triangular face to the midpoint of a base edge. They are related by ℓ = √(h² + (a/2)²), so ℓ is always the longer of the two. Plugging a slant height into the volume formula is the classic error — it overstates volume. Pick the slant-height mode and this page converts first.
How it’s calculated
V = a²h/3, with a the base edge and h the vertical height. Slant-height mode first converts via h = √(ℓ² − (a/2)²); the reverse relation ℓ = √(h² + (a/2)²) is reported either way. Values rounded to 3 decimals.
Assumes a right pyramid — apex directly above the base center. The volume formula survives an off-center apex (same h), but the slant-height relations do not.
Real square pyramids through the formula
| Pyramid | Base edge × height | Volume (a²h/3) |
|---|---|---|
| Great Pyramid of Giza (original) | 230.3 m × 146.6 m | ≈ 2.59 million m³ |
| Pyramid of Khafre | 215.3 m × 136.4 m | ≈ 2.11 million m³ |
| Luxor hotel, Las Vegas | 183 m × 106.7 m | ≈ 1.19 million m³ |
Computed with V = a²h/3 from commonly published dimensions; rounded to three significant figures.
Common mistakes
- Plugging the slant height into V = a²h/3 — volume needs the vertical height.
- Using the full base edge instead of a/2 when converting between slant and vertical height.
- Forgetting to square the base edge — the base area is a², not a.
- Dividing by 2 instead of 3; a pyramid is a third of its box, not half.
Frequently asked questions
What is the volume formula for a square pyramid?
V = a²h/3: base edge squared, times the vertical height, divided by 3. With a = 6 and h = 8 that is 36 × 8 / 3 = 96 cubic units.
What is the difference between height and slant height?
Height h is vertical, apex to base center. Slant height ℓ runs down the middle of a face to a base edge, so ℓ = √(h² + (a/2)²) and is always longer. Volume uses h; face area uses ℓ.
Why divide by 3?
Three congruent pyramids can be assembled into the full box with the same base and height — a dissection known since antiquity and confirmed by integration. The 1/3 applies to every pyramid and cone, whatever the base shape.
I only know the lateral edge (apex to a corner). Can I still get volume?
Yes. The lateral edge e satisfies e² = h² + a²/2, so compute h = √(e² − a²/2) first, then use vertical-height mode.