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 — hexagonal pavilion roof shingles: 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” (100 sq ft = 9.29 m²). You need about 2.24 squares — buy 2.5 squares (225 sq ft) for waste and ridge caps where the triangles meet.

Worked example — hexagonal church spire cladding: 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’s a lot of vertical surface for a relatively small footprint — typical of tall spires.

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 is from the MIDPOINT of a base edge to the apex — NOT from a corner.

For a hexagon with edge a, the apothem (distance from center to edge midpoint) is a√3/2. The vertex distance (center to corner) is a. These are different by about 13.4%.

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 or h = 0: SA degenerates (no shape). ✓
• For a = 1, h = √3/2 (regular hex pyramid where all 12 edges equal): SA = (3√3/2) + 3 × √3 = 4.5 × √3 ≈ 7.79.