Right Rectangular Pyramid Calculator

Background.

A pyramid holds exactly one third of the box that would enclose it. Take a prism with the same base and the same height and the pyramid inside fills a third of it, however tall or squat the shape. The fraction is easiest to believe with a cube: six identical square pyramids, each with one face of the cube as its base and the cube's centre as its apex, fill it completely. Each is one sixth of the cube, which is exactly one third of its base area times its height of half a cube.

The page handles a right pyramid with a square base, the shape of the Great Pyramid at Giza, a square hip roof, or a tent pitched around a centre pole. Right means the apex sits directly above the centre of the base, so all four triangular faces are identical. Despite the name, the base has to be square: there is a single side length to enter, and a pyramid on a long, narrow rectangle needs a separate length and width that the page cannot take.

Two lengths fix everything else. The base side gives the base area, side squared. The perpendicular height, measured straight down from the apex to the base rather than along a face, gives the volume. The slant height, the height of each triangular face from the middle of a base edge up to the apex, follows from Pythagoras, because the vertical height, half the base side and the slant height form a right triangle inside the solid.

The slant height is the figure you need for the outside surface, whether for roofing, cladding or cutting card for a model. Four triangular faces of base s and height l have a combined area of 2 × s × l.

What is right rectangular pyramid calculator?

A right square pyramid has a square base and four congruent isosceles triangles meeting at an apex directly above the centre of that base. Its volume is one third of the base area multiplied by the perpendicular height. Its slant height is the altitude of each triangular face, measured from the midpoint of a base edge to the apex, and is shorter than the lateral edge that runs from a base corner to the apex. Results come out in the unit you enter, squared for area and cubed for volume.

How to use this calculator.

1. 01Enter the Square base side, the length of one edge of the base. Any unit works, provided the height uses the same one.
2. 02Enter the Perpendicular height, measured vertically from the apex to the centre of the base. If you only know the slant height l, the height is √(l² − (s ÷ 2)²).
3. 03Read the volume in cubic units, the base area in square units and the face slant height in the length unit you entered.
4. 04For the outside surface, multiply 2 × base side × slant height to get the four faces, and add the base area if the bottom is covered too.

The formula.

The base area is A = s², where s is the base side, and the volume is V = A × h ÷ 3 = s² × h ÷ 3, where h is the perpendicular height. The factor of one third holds for every pyramid and cone, whatever the shape of the base, because horizontal cross-sections shrink with the square of their distance from the apex. For the slant height the page uses a right triangle inside the pyramid: one leg is the height h, the other is the base apothem, the distance from the centre of the base to the middle of an edge, which for a square is s ÷ 2. The hypotenuse is the slant height, l = √(h² + (s ÷ 2)²). Volume grows with the square of the base side, so doubling s quadruples V, while doubling h only doubles it.

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.