This pyramid calculator finds the volume, slant height and total surface area of a square-based pyramid from its base side and vertical height — the Great Pyramid proportions included as the default.
Slant height here is the triangular face’s height (apex to the midpoint of a base edge) — not the pyramid’s corner edge, which is longer (√(h² + (s/√2)²)). Surface discussions that mix the two inflate by ~15% on typical proportions.
One Third of the Box It Sits In
V = ⅓ × base area × height — the same tapering law as our Cone Calculator, for polygonal bases. The face slant ℓ = √(h² + (s/2)²) gives lateral area = 2sℓ (four triangles of ½·s·ℓ), and total area adds the base: s² + 2sℓ. Part of our Math Calculators collection.
Worked Example (Verified)
- Base 6, height 8 (defaults): V = 36 × 8 ÷ 3 = 96 · ℓ = √(64 + 9) = 8.54 · total area = 36 + 2 × 6 × 8.54 ≈ 138.53
- Giza check (base ≈ 230.4 m, original height ≈ 146.6 m): the formula gives ≈ 2.59 million m³ — matching the surveyed volume of the Great Pyramid, casing stones and all, 4,500 years on
- The ⅓ in daily life: a pyramid-shaped pile (grain, sand, gravel) holds a third of the box you’d naively fence around it — ordering three piles’ worth of truck space for one pile is the classic rookie surplus
- Square vs triangular faces: this tool is the square pyramid (four triangular faces); a tetrahedron is the triangular-based cousin with its own formulas
Shape Notes
- Regular vs irregular: these formulas assume the apex sits over the base center (right pyramid) — lean the apex and volume survives (Cavalieri) but surface area gets bespoke.
- The golden-ratio romance about Giza (slant/height ratios ≈ φ) is a numerological rabbit hole; the Egyptians worked in seked (slope) units and the ratio falls out of a 5½-palm seked.
- Frustum of a pyramid (flat top) — volume via the prismoidal extension — appears in everything from buckets to Mesoamerican platforms.
Related Tools
- Cone Calculator — the round-based sibling sharing the ⅓ law
- Volume Calculator — the wider solids shelf
- Math Calculators — the full collection
Frequently Asked Questions
How do you find the volume of a pyramid?
V = ⅓ × base area × vertical height. For a square base: ⅓ s² h. Base 6, height 8 → 96 cubic units. The ⅓ is universal for pyramids and cones — any tapered solid to a point holds a third of its enclosing prism.
What is the slant height of a pyramid?
The height of one triangular face, measured from apex down to the midpoint of a base edge: ℓ = √(h² + (s/2)²) for a square pyramid. It is not the corner edge (which runs to a vertex and is longer) — surface-area formulas want the face slant.
How do you find a pyramid’s surface area?
Base area + four triangular faces: s² + 4 × (½ s ℓ) = s² + 2sℓ. For base 6 and slant 8.54, that’s 36 + 102.53 ≈ 138.53 square units. Leave the base term out for an open-bottom (roof-style) structure.
Does the ⅓ rule work for any pyramid shape?
Yes — triangular, square, hexagonal, irregular bases: V = ⅓ × (base area) × (vertical height), provided height is measured perpendicular to the base plane. It’s Cavalieri’s principle: cross-sections shrink quadratically toward the apex.
How big is the Great Pyramid by these formulas?
With base ≈ 230.4 m and original height ≈ 146.6 m: volume ≈ 2.59 million m³ and (with slant ≈ 186 m) a cased surface of ≈ 224,000 m² — numbers the calculator reproduces from just the two inputs.
Is this pyramid calculator free?
Yes — free, instant, no sign-up.