Hexagonal Pyramid Surface Area Calculator

A regular hexagonal pyramid has a hexagonal base and six congruent isosceles triangular side faces meeting at the apex.

SA = (3√3 / 2) × a² + 3 × a × l

Where:

• a = hexagonal base edge
• h = perpendicular height (base center to apex)
• l = slant height of each triangular face (from base edge midpoint to apex)

The slant l is computed from h and the hexagon’s apothem (the apothem of a regular hexagon = a√3/2):

l = √(h² + (a√3/2)²) = √(h² + 3a²/4)

The first term in SA is the hexagonal base; the second is the six triangular sides combined (each side has area ½ × a × l, so 6 sides = 3al).

Worked example: shingling a hexagonal pavilion roof A six-sided garden pavilion with floor edge a = 2 m and roof apex h = 3 m above the floor. Apothem of hexagonal base: 2 × √3/2 = √3 ≈ 1.732 m. Slant height: l = √(9 + 3) = √12 ≈ 3.464 m.

Hexagonal floor area (if covered): 2.598 × 4 ≈ 10.39 m². Usually the floor doesn’t need shingles. Six triangular roof panels: 3 × 2 × 3.464 ≈ 20.78 m² of roof surface.

Roof shingles are sold by “the square”, which is 100 sq ft or 9.29 m². This roof is 20.78 / 9.29 = 2.24 squares. Buy 2.5 squares, meaning 250 sq ft, and use the extra on waste and on the six hip lines where the triangles meet. Hip caps eat shingles fast on a six-sided roof, because there are six hips on a plan area that a square roof would cover with four.

Worked example: cladding a hexagonal church spire A church steeple with hexagonal cross-section narrowing to a point: a = 1 m, h = 8 m (tall and narrow). Slant: l = √(64 + 0.75) ≈ 8.046 m. Six triangular panels: 3 × 1 × 8.046 ≈ 24.14 m² of copper or slate cladding.

That is 24 m² of cladding standing over a footprint of just 2.6 m², which is what makes spires expensive to re-roof relative to their plan area.

Where hexagonal pyramid surface matters:

• Pavilion and gazebo roof shingles. Six-sided pavilions are common, and the roof is exactly this shape.
• Tower spire cladding. Castle towers and church steeples often have hexagonal pyramidal roofs.
• Decorative finial fabrication. Hexagonal pointed caps for fence posts, garden statuary.
• Custom jewelry. Hex-pyramid pendants and rings.
• Pencil tip after sharpening. The wood-and-lead exposed pyramid surface area (rarely calculated in practice).

Slant height, the recurring gotcha:

The slant height l runs from the MIDPOINT of a base edge to the apex, never from a corner.

For a hexagon with edge a, the apothem (centre to edge midpoint) is a√3/2. The vertex distance (centre to corner) is a. The corner sits 15.5% further out than the edge midpoint does, and that difference carries all the way up to the apex.

Use apothem for slant height calculations (it’s perpendicular to the base edge). Use vertex distance only for edge-length calculations (apex-to-corner edges of the pyramid).

Lateral only vs. total:

• Solid hex pyramid (decorative, with floor): include base. SA = (3√3/2)a² + 3al.
• Open hex pyramid (roof, tent fly): lateral only. SA = 3al.

For most roofing applications, just use 3al.

Sanity check:

• a = 0: SA = 0. ✓
• h = 0: the solid flattens and SA collapses to twice the hexagon, 3√3 × a². ✓
• For a = 1, h = √3/2: the apothem is also √3/2, so l = √(3/4 + 3/4) = √1.5 ≈ 1.2247. Lateral = 3 × 1.2247 ≈ 3.674, base = 2.598, total ≈ 6.272. ✓

One shape you cannot build. Ask for a hexagonal pyramid whose twelve edges are all equal and the algebra refuses. The apex-to-corner edge is √(h² + a²), and setting that equal to a forces h = 0. Six equilateral triangles meeting at a point already fill the full 360° and lie flat, with no angle left over to lift into a peak. That is why the square pyramid has an equilateral-faced version and the hexagonal one does not, and it is the same arithmetic that stops hexagons alone from closing into a ball.