Height Of A Square Pyramid Calculator

Background.

A square pyramid has three different "heights", and mixing them up is the commonest mistake in pyramid problems. The true height runs straight down from the apex to the centre of the base, at right angles to it. The slant height runs down the middle of one triangular face, from the apex to the midpoint of a base edge. The lateral edge runs from the apex to a corner. They always come in that order, shortest to longest, and only the first belongs in the volume formula.

Here you supply the side of the square base and the perpendicular height, and the page returns the base area, the volume and the slant height. Volume is one third of base area times height, exactly a third of the box that would just enclose the pyramid. The slant height comes from a right triangle hidden inside the solid: its vertical leg is the height, its horizontal leg is half the base side (the distance from the centre of the square to the middle of an edge), and its hypotenuse lies along the face.

In practice people often know the slant height or the volume and need the vertical height instead. The same relationships run backwards: the height is the square root of the slant height squared minus half the side squared, or three times the volume divided by the base area. Work it out that way first, then enter it to get the remaining measurements.

The pyramid is assumed to be right and regular, with its apex directly above the centre of a square base. Shift the apex off-centre and the volume formula still holds, but the four faces no longer share one slant height, so that output would stop meaning anything.

What is height of a square pyramid calculator?

The height of a square pyramid is the perpendicular distance from the apex to the plane of the base. In a right pyramid the apex sits directly over the centre of the square, so the height meets the base exactly at its centre. It is shorter than both the slant height, measured along the middle of a face, and the lateral edge, measured from apex to corner. Together with the base side, it fixes the pyramid's volume, its face slant height and every other dimension of a regular square pyramid.

How to use this calculator.

1. 01Enter the length of one edge of the square base in Square base side. All four edges are equal on a regular pyramid; if yours differ, this page does not apply.
2. 02Enter the vertical height in Perpendicular height, apex to the centre of the base at right angles to it. If you only know the slant height or the volume, convert it to the vertical height first, as the questions below show.
3. 03Use one length unit for both fields. Volume then comes out in that unit cubed, base area in that unit squared, and slant height in the unit itself.
4. 04Use the slant height for anything that covers the faces, such as cladding, fabric or a paper net, and the volume for whatever fills the inside.

The formula.

With base side s and perpendicular height h: base area A = s², volume V = A × h ÷ 3 = s²h ÷ 3, and slant height ℓ = √(h² + (s/2)²). The one-third is the factor every pyramid and cone shares: a pyramid holds exactly a third of the prism with the same base and height. A cube shows it neatly, splitting into three identical square pyramids that share one corner as their apex. In the slant-height formula, s/2 is the base apothem, the distance from the centre of the square to the midpoint of a side; with h it forms the two legs of a right triangle whose hypotenuse runs down the face. Volume grows with the square of the base side but only in proportion to the height, so a 10% wider base adds 21% to the volume while a 10% taller pyramid adds 10%.

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.