Square Pyramid Calculator

Background.

A square pyramid — four triangular faces meeting at an apex over a square base — is governed by one ancient and initially surprising rule: its volume is exactly one-third of the box that would enclose it. Base side s and perpendicular height h give V = s²h/3, and this page returns that volume together with the base area and the faces' slant height.

The one-third is not an approximation. Three identical oblique pyramids assemble exactly into a cube — a dissection the ancient Egyptians' successors and Euclid both knew — and calculus confirms it in a line: stacking square cross-sections that shrink quadratically toward the apex integrates to a third of base times height. It is the same one-third that appears in cones (circular pyramids) and every other apex-over-a-base solid.

The slant height is the practical second output. The height h runs invisibly up the middle; what a roof panel, a face of sheet metal, or a coat of paint follows is the slant height ℓ up the centre of each triangular face, and the two are related through the base's half-side: ℓ = √(h² + (s/2)²). Confusing the two is the classic pyramid error — slant is always the longer — and lateral surface area hangs off the slant, not the height: each face has area sℓ/2, four faces total 2sℓ.

Pyramid arithmetic is far from ornamental: pyramidal hoppers and roof sections, concrete footings poured as truncated pyramids, and the eternal comparison of monument volumes (the Great Pyramid's ≈230 m base and 147 m original height compute to ≈2.6 million m³ by exactly this formula) all run through V = s²h/3. The model here is the right, regular case — apex over the base's centre — as the scope note records.

What is square pyramid calculator?

A square pyramid is the solid with a square base of side s and four triangular faces rising to a single apex; this page treats the right, regular case with the apex directly above the base's centre at perpendicular height h. Its three standard quantities: volume V = s²h/3 (one-third of the enclosing box — the universal pyramid rule), base area s², and face slant height ℓ = √(h² + (s/2)²), the distance from a base edge's midpoint up the face to the apex — the length that governs roofing, cladding, and lateral surface area (2sℓ).

How to use this calculator.

1. 01Enter the base side and the perpendicular height — the vertical drop from apex to base centre, not the sloped edge or face length.
2. 02Read the volume (s²h/3), base area (s²), and face slant height ℓ.
3. 03For material take-offs, build lateral surface from the slant: each triangular face is s×ℓ/2, so four faces total 2sℓ — add the base s² only if that surface is real (a roof has no floor).
4. 04Given a slant measurement instead of height — common when only the outside is accessible — recover h = √(ℓ² − (s/2)²) first, then enter it.
5. 05For hoppers and truncated shapes, compute as the big pyramid minus the small one cut off — both by this page's formula — rather than reaching for a slab approximation.

The formula.

Volume: slice the pyramid horizontally at fraction t of the way down from the apex and the cross-section is a square of side t·s — linear shrinkage — so its area is t²s², and integrating t² from 0 to 1 yields the 1/3: V = s²h/3. The dissection proof says the same without calculus: three congruent pyramids, apexes at one corner, tile a cube exactly. Consequences follow from the squared term — doubling the base side quadruples volume while doubling height merely doubles it, and half the pyramid's volume lies in the bottom 20.6% of its height, which is why hoppers drain slowly at the end. Slant height: drop the apex to the base centre (length h), walk half a side (s/2) to a base edge's midpoint, and rise up the face; the first two are the legs of a right triangle whose hypotenuse is ℓ = √(h² + (s/2)²). A third length — the lateral edge from apex to a base corner — uses the half-diagonal instead: √(h² + s²/2), longer again; keeping the three straight is most of pyramid competence. The engine evaluates all three outputs in Decimal arithmetic, rounding once to twelve significant digits.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax, Rice University. Precalculus 2e, 2021. Trigonometry and analytic geometry chapters. Retrieved 2026-08-06. independence: primary; access: open.
2. [2]OpenStax, Rice University. College Algebra 2e, 2021. Algebra and function chapters. Retrieved 2026-08-06. independence: secondary-check; access: open.
3. [3]NIST/SEMATECH. e-Handbook of Statistical Methods, 2012. Chapter 1.3 and Chapter 4. Retrieved 2026-08-06. independence: primary; access: open.