Hexagonal Pyramid Calculator

A regular hexagonal pyramid sits on a regular hexagonal base and tapers to a point (apex) directly above the center of the base. Six triangular faces connect the base to the apex.

The base area is the same as any regular hexagon: A = (3√3/2) × a². With base area and height, volume follows from the standard pyramid formula, one-third of base times height:

Volume = (1/3) × (3√3/2) × a² × h = (√3/2) × a² × h

The slant height l is the distance from the apex down to the midpoint of any base edge. That midpoint is the apothem away from the center, so:

Apothem = a × √3 / 2 Slant height l = √(h² + apothem²) = √(h² + 3a²/4)

Each triangular face has base a and height l (the slant height), giving 6 × (1/2 × a × l) = 3al for lateral surface area.

Lateral SA = 3al Total SA = 3al + (3√3/2) × a²

Worth noting: the slant height runs to the middle of a base edge, not to a corner. The edge from apex to corner is longer, at √(h² + a²), because a regular hexagon’s distance from centre to corner is exactly a. Textbooks confuse the two constantly, and using the corner edge in place of the slant height inflates the surface area every time.

That same fact rules out something people often ask for. A hexagonal pyramid with all twelve edges equal would need √(h² + a²) = a, which forces h = 0 and flattens the solid into a plate. There is no such thing as a hexagonal pyramid with equilateral triangular faces. Six equilateral triangles around a point close up flat, with nothing left over to lift into a peak, which is the same 360° arithmetic that stops hexagons from folding into a ball.

Worked example: a six-sided gazebo roof

A gazebo with 2 m floor edges and an apex 3 m above the plate. Apothem = 2 × √3/2 = 1.732 m. Slant height = √(9 + 3) = 3.464 m. Lateral surface = 3 × 2 × 3.464 = 20.78 m² of roof to shingle. Enclosed volume = (√3/2) × 4 × 3 = 10.39 m³ of attic space under the peak. Rafter length, apex to corner, = √(9 + 4) = 3.606 m, so order six at 3.7 m and trim.

Notice that the rafter runs 4% longer than the slant height. On a small roof that is 14 cm, which is the difference between a rafter that reaches the corner and one that does not.

Hexagonal pyramids appear in architecture (temple spires, decorative finials, gazebo roofs), in crystallography, where apatite and beryl terminate this way, and in geometry courses as the six-sided member of the pyramid family.