Square Pyramid Volume Calculator
Compute the volume of a square-based pyramid from base side and height.
For Egyptian-style pyramid layouts, tent volumes, and roof attics.
V = (1/3) × a² × h
Where a is the base edge (one side of the square at the bottom) and h is the vertical perpendicular height from base to apex (NOT a slant edge).
A square pyramid has a square base and four triangular sides meeting at a point on top — like the Great Pyramid of Giza, a wizard hat with a square brim, or a pavilion roof.
Worked example — Great Pyramid of Giza: Original dimensions: base ≈ 230.4 m square, original height ≈ 146.6 m (now 138.8 m due to erosion). V = (1/3) × 53,084 × 146.6 ≈ 2,593,000 m³ of stone. At an average block volume of 1.5 m³, that’s around 1.73 million blocks — close to estimates of 2.3 million (the difference is interior chambers and some quarrying losses).
Where square pyramids show up:
- Egyptian-style pyramids. Giza, Khufu, Khafre — all near-perfect square pyramids.
- Modern architectural pyramids. The Louvre entrance pyramid is 35 m square at the base, 21.6 m tall — interior volume ≈ 8,820 m³.
- Camping tents. A 4-pole pyramid tent (no center pole) approximates a square pyramid. Helps with airflow calculations.
- Hipped roof attics. A perfectly hipped roof over a square house is exactly a square pyramid. Useful for attic insulation and ventilation sizing.
- Pavilion roofs. Gazebo and pavilion roofs are often square pyramidal.
- Pyramid-shaped tea bags. The Tetley/Lipton brand pyramid bags are tetrahedra, not square pyramids — a different shape entirely.
Pyramid vs. prism — the 1/3 rule:
Every pyramid (with any flat base) is exactly 1/3 the volume of the prism with the same base and height. This is why the formula has the (1/3) factor — you’re filling in 1/3 of a rectangular box. The pyramid tapers smoothly to zero at the top, so the average cross-section is 1/3 of the base.
The “height” gotcha — perpendicular vs. slant:
There are at least three different “heights” people confuse:
- h (perpendicular height): the vertical distance from base center to apex. Use this in the volume formula.
- slant height (l): the distance from base edge midpoint to apex, along the triangular face. Use for surface area, not volume.
- edge length: the distance from base corner to apex. Different from both above.
For a 100×100 m base with apex 50 m straight up: h = 50, slant height = √(50² + 50²) = 70.7, edge length = √(50² + 70.7²) = 86.6. Three different numbers; only h goes into the volume formula.
Sanity check:
- a = 0 or h = 0: V = 0. ✓
- For a square pyramid with a = h = 1: V = 1/3. (1/3 of the bounding 1×1×1 cube.)