Right Square Pyramid Calculator

Background.

A pyramid on a square base holds exactly one third of the box that would just enclose it. Greek geometers knew this long before algebra: Archimedes credited Democritus with the discovery and Eudoxus with the proof. It is the whole of the volume rule, V = s² × h ÷ 3, where s is the side of the base and h the height.

Those two lengths fix a right square pyramid completely. "Right" means the apex sits directly above the centre of the square, so the four triangular faces are identical and the height is measured straight down from the tip to that centre, not along a face.

The other length people usually need is the slant height: the distance from the apex down the middle of a face to the midpoint of a base edge. It is the height of each triangular face, so it is what you use for the area of the sides, the length of material running up the centre of each face of a pyramid roof, or the fold line on a paper model. It comes from a right triangle hidden inside the pyramid, with the vertical height and half the base side as its legs.

Do not confuse it with the lateral edge, the sloping line from the apex to a corner. A corner is farther from the centre than an edge midpoint, so the edge is longer: √(h² + s²/2) rather than √(h² + s²/4).

Any length unit works as long as both inputs use it. The Great Pyramid of Giza, about 230.3 m on each side and about 146.6 m tall when complete, comes to roughly 2.59 million cubic metres.

What is right square pyramid calculator?

A right square pyramid has a square base and four congruent isosceles triangular faces meeting at an apex directly above the centre of the base. Its size is set by two lengths, the base side s and the perpendicular height h. From them follow the base area s², the volume s² × h ÷ 3, and the slant height √(h² + (s/2)²), which is the height of each triangular face measured along its centre line.

How to use this calculator.

1. 01Enter the length of one side of the square base.
2. 02Enter the perpendicular height, from the apex straight down to the centre of the base. If you can only measure up the middle of a face, that is the slant height l; the height is √(l² − (s/2)²).
3. 03Use the same unit for both. The base area comes out in that unit squared, the volume in that unit cubed and the slant height in the unit itself.
4. 04For the area of the four sloping faces, multiply the slant height by twice the base side; add the base area if the bottom counts too.

The formula.

Base area B = s². Volume V = B × h ÷ 3 = s² × h ÷ 3. Slant height l = √(h² + (s/2)²), where s/2 is the apothem of the square base, the distance from its centre to the midpoint of a side. The one-third has a neat demonstration: six identical pyramids with their apexes at the centre of a cube of side s fill it exactly, each with base s² and height s/2, and 6 × s² × (s/2) ÷ 3 = s³. Volume grows with the square of the base side but only in proportion to the height, so a base 10% wider adds 21% to the volume, while a pyramid 10% taller gains just 10%. Slant height always exceeds both h and s/2, since it is the hypotenuse of the triangle they form.

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.
2. [2]OpenStax, Rice University. College Algebra 2e, 2021. Algebra and function chapters. Retrieved 2026-08-06.
3. [3]NIST/SEMATECH. e-Handbook of Statistical Methods, 2012. Chapter 1.3 and Chapter 4. Retrieved 2026-08-06.