Square Pyramid Volume Calculator

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. Think of the Great Pyramid of Giza, a wizard hat with a square brim, or a pavilion roof.

Worked example, the Great Pyramid of Giza Original dimensions: base about 230.4 m square, original height about 146.6 m (now 138.8 m after erosion). V = (1/3) × 53,084 × 146.6 ≈ 2,594,000 m³ of stone. At an average block volume of 1.5 m³ that comes to roughly 1.73 million blocks, which is in the same range as the usual estimate of about 2.3 million. The gap is real, not a rounding error: the core blocks are not all 1.5 m³, the interior chambers and galleries are voids rather than stone, and mortar fills a surprising amount of the joint space.

Where square pyramids show up

• Egyptian-style pyramids. The Khufu, Khafre and Menkaure pyramids at Giza are all near-perfect square pyramids.
• Modern architectural pyramids. The Louvre entrance pyramid is 35 m square at the base and 21.6 m tall, so the interior volume is about 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 and Lipton pyramid bags are tetrahedra, not square pyramids. Different shape, different formula.

Pyramid against prism: the 1/3 rule

Every pyramid, with any flat base, is exactly 1/3 the volume of the prism sharing that base and height. That is where the (1/3) comes from. You are filling one third 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, slant, and edge

Three different lengths get called “the height”, and they are not interchangeable:

• h (perpendicular height): the vertical distance from base center to apex. This is the one the volume formula wants.
• slant height (l): base edge midpoint to apex, measured up the middle of a triangular face. Surface area uses it. Volume does not.
• lateral edge (e): base corner to apex, along the arris where two faces meet. Longer than both.

For a 100 m square base with the apex 50 m straight up: h = 50, slant height = √(50² + 50²) = 70.7, lateral edge = √(50² + 100²/2) = √7,500 = 86.6. Three different numbers, and only h belongs in the volume formula.

The edge formula is the one people get wrong, because the half-base has to appear twice, once for each horizontal direction: e = √(h² + a²/2), not √(h² + (a/2)²). Those two happen to agree when h equals a/2, which is exactly the case in the example above, so do not learn the formula off that line.

Sanity check

• a = 0 or h = 0: V = 0. ✓
• a = h = 1: V = 1/3, one third of the bounding 1 × 1 × 1 cube. ✓
• Double the base edge and the volume goes up four times, not two. Base area is the square of a.