How this calculator works
A regular hexagonal base splits into six equilateral triangles. Its apothem is √3 times half the base edge. Combine that apothem with the vertical height to find a triangular face’s slant height. Six matching face areas give the lateral area, and one base gives the total. Horizontal sections remain similar hexagons: their linear size falls in proportion to the distance remaining to the apex, while their area falls with its square. Reverse modes recover height from volume or face areas, or recover the edge from height and volume.
Formula
Base B = (3√3/2)a²; apothem r = √3a/2; slant s = sqrt(h²+r²); face F = as/2; lateral L = 6F = 3as; total S = B+L; volume V = Bh/3. At height z, section area = B(1-z/h)².
Worked example
With base edge 2 m and height 1 m, the apothem is √3 m and the slant is 2 m. Each face is 2 m², lateral area is 12 m², base area is 6√3 m² and volume is 2√3 m³. Halfway up, the section area is one quarter of the base area.
Advertisement
Frequently asked questions
Is a hexagonal pyramid the same as a hexagonal prism?
No. The pyramid narrows to one apex. With equal base and height it has one third of the prism’s volume.
Are all six triangular faces equilateral?
They are congruent isosceles triangles, but they are equilateral only at a particular height. The hexagon’s six central base triangles are equilateral.
Why is a halfway section only one quarter of the base area?
Its side length is half as large, and area scales with the square of that factor: (1/2)²=1/4.