Hexagonal Pyramid Calculator

Background.

Six triangles leaning in from the edges of a regular hexagon to meet at a single point: that is a hexagonal pyramid, the shape of many gazebo and pavilion roofs, of some tent tops and, roughly, of the tip of a quartz crystal. Working with one usually means answering one of three questions. How much does it hold? How large is its footprint? And how long is each face from base to apex, which is the number you need to cut roofing panels or fabric?

All three follow from two measurements, the side of the hexagon and the perpendicular height. A regular hexagon splits into six equilateral triangles, so its area is six times (√3/4)s², or about 2.598 times the side squared. The volume is one third of that area times the height, exactly a third of a hexagonal prism with the same base and height, the same one-third rule that holds for every pyramid and cone. The slant height of each face comes from a right triangle standing inside the solid, with the height as one leg and the base apothem, the distance from the hexagon's centre to the middle of a side, as the other.

Height is the measurement most often got wrong. It is the straight vertical distance from the base to the apex, not the length along a face or an edge. If you can only measure along a face, subtract the apothem squared from the slant height squared and take the square root.

The pyramid is assumed to be right and regular: six equal sides and the apex directly above the centre. Results carry no units, so enter both lengths in the same unit and read the area in that unit squared and the volume in that unit cubed.

What is hexagonal pyramid calculator?

A hexagonal pyramid is a solid with a hexagonal base and six triangular faces meeting at an apex; it has seven faces, twelve edges and seven vertices. In a right regular hexagonal pyramid the base is a regular hexagon and the apex sits directly above its centre, so all six faces are congruent isosceles triangles. Its size is fixed completely by the base side and the perpendicular height.

How to use this calculator.

1. 01Enter the Regular hexagon side, the length of one edge of the base. The distance across opposite corners of a regular hexagon is twice the side, so a base 3 m from corner to corner has 1.5 m sides.
2. 02Enter the Perpendicular height, measured straight up from the base to the apex. If you only know the slant height l of a face, the height is √(l² − 0.75 × side²).
3. 03Keep both lengths in the same unit. The base area comes out in that unit squared and the volume in that unit cubed.
4. 04For roofing or a fabric cover, multiply the slant height by three times the side to get the combined area of the six faces.

The formula.

Base area A = (3√3 ÷ 2) × s², about 2.598 × s², because a regular hexagon of side s is six equilateral triangles, each of area (√3 ÷ 4) × s². Volume V = A × h ÷ 3, the general rule for any pyramid. The base apothem a = (√3 ÷ 2) × s is the distance from the centre of the hexagon to the midpoint of a side. The face slant height l = √(h² + a²) by Pythagoras, because the height, the apothem and the slant height form a right triangle inside the pyramid. Since a² = 0.75 × s², the slant height can also be written √(h² + 0.75 s²). Volume grows with the square of the side but only in proportion to the height, so doubling the side quadruples the volume while doubling the height merely 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.