How this instrument works
A hexagonal pyramid is a regular hexagon base topped by six triangular faces that all meet at one apex. Its total surface area splits cleanly into two pieces: the flat base plus the six sloped triangles, totalArea = baseArea + lateralArea. The base piece, (3√3 ⁄ 2)s², is the same regular-hexagon formula this batch's own hexagon page reports, and it falls out of a neat six-fold split: a regular hexagon is the only regular polygon whose side length equals the distance from its centre to a corner, so it divides exactly into six equilateral triangles of side s. Each of those has area (√3⁄4)s², and six of them sum to (3√3⁄2)s² — no separate derivation needed once that hexagon page's own result is on hand.
The lateral piece leans on the same six-fold symmetry from a different direction. Because the apex sits directly above the hexagon's centre and every base edge has the same length s, the six triangular side faces are congruent, each with base s and height equal to the slant height. One face has area ½·s·slant, and stacking six of those halves collapses to the tidy shortcut lateralArea = 3·s·slant — a simplification that only holds because the base is regular; tilt the apex off-centre or stretch one side and each face needs its own separate calculation instead.
Slant height here means the distance from the apex straight down the centre of one triangular face to the midpoint of its base edge — not the lateral edge running from apex to a corner, which is longer for any pyramid taller than it is wide. Push the slant height toward zero and the six faces flatten away entirely, leaving only the bare hexagonal base; push the side length toward zero instead and the whole base area vanishes along with the lateral area, since every triangular face shrinks to nothing alongside it.
- Enter the pyramid's Base side length in the s field — one edge of the regular hexagon forming the bottom.
- Enter Slant height — the distance from the apex down the middle of a triangular face to the midpoint of its base edge, not the corner-to-corner lateral edge.
- Read Base area for the hexagon alone, computed as (3√3 ⁄ 2)s².
- Read Lateral area (6 triangles) for the six sloped faces combined, and Total surface area for base plus lateral together.
- Change either input and all three outputs recompute together, so a mismatched slant height shows up immediately.
Worked example — side 4, slant height 6
Take a hexagonal pyramid with Base side length s = 4 and Slant height = 6 units. The base area follows the six-equilateral-triangle split: baseArea = (3√3 ⁄ 2) × 4² = 24√3 ≈ 41.569219381653056 square units, six triangles of side 4 fanned out from the hexagon's own centre.
Each of the six triangular faces has area ½ × 4 × 6 = 12, and six of them combine through the 3·s·slant shortcut to lateralArea = 3 × 4 × 6 = 72 square units exactly. Total surface area sums the two pieces: totalArea = 41.569219381653056 + 72 = 113.56921938165306 square units, the figure to quote when the whole solid, base included, needs covering.
Questions
What is the formula for a hexagonal pyramid's surface area?
Total surface area is baseArea + lateralArea, where baseArea = (3√3 ⁄ 2)s² covers the regular hexagonal base and lateralArea = 3·s·slant covers its six triangular side faces combined. With side length 4 and slant height 6 that gives 24√3 + 72 ≈ 113.569 square units, and this sheet reports both pieces separately as well as the sum.
Why does the lateral area formula come out to 3·s·slant?
Because the six triangular faces of a regular hexagonal pyramid are congruent — same base s, same slant height — each contributes ½·s·slant, and six of those halves collapse to 3·s·slant. That shortcut only holds because the apex sits directly above the hexagon's centre; an off-centre apex or an uneven base needs each face measured on its own.
What's the difference between slant height and the pyramid's lateral edge?
Slant height runs from the apex straight down the middle of one triangular face to the midpoint of a base edge — the height used inside each face's own area formula. The lateral edge instead runs from apex to a base vertex, a corner rather than an edge midpoint, and it is the longer of the two for any pyramid taller than it is wide. Entering the edge length where slant height belongs overstates every triangular face.
How does the base area relate to a regular hexagon's own area formula?
It is the identical formula — a hexagonal pyramid's base is a plain regular hexagon, so baseArea uses exactly the (3√3 ⁄ 2)s² result this batch's hexagon page reports on its own, itself six equilateral triangles of side s fanned out from the centre. Standing the hexagon up under an apex changes nothing about how its own footprint is measured.
What happens to the surface area when the slant height is zero?
The pyramid flattens to the bare hexagonal base: lateralArea drops to zero, since 3·s·0 = 0, and totalArea equals baseArea alone — 41.569219381653056 square units for a side length of 4. With no slant height there are no real triangular faces left standing, only a flat hexagon lying on the table.
Does this formula work for an irregular hexagonal base?
No — both baseArea = (3√3 ⁄ 2)s² and lateralArea = 3·s·slant assume a regular hexagon with six equal sides, a single shared slant height, and an apex centred directly above it. An irregular base needs each of the six triangular faces measured and summed on its own, since their base lengths and slant heights would all differ.