Quick answer: V = ⅓ × (3√3/2)a² × h, which simplifies to V = (√3/2)a²h ≈ 0.866 a²h. A hexagonal pyramid with side 6 and height 8 holds 249.42 cubic units. For surface area you need the slant height, built from the apothem — not the side.
The formula, and the simplification worth knowing
Every pyramid follows the same rule: one third of the base area times the vertical height. For a hexagonal base that means:
| Step | Expression | Value |
|---|---|---|
| Hexagonal base area | (3√3/2)a² ≈ 2.598a² | 93.5307 |
| Times the height | 2.598a²h | 748.2459 |
| Times one third | (√3/2)a²h ≈ 0.866a²h | 249.4153 |
Notice what happens to the coefficient: (1/3) × (3√3/2) = √3/2. The 3 cancels cleanly, leaving 0.866 — a much friendlier number than 2.598, and one worth committing to memory if you do this often.
Why one third, always. Any pyramid occupies exactly one third of the prism sharing its base and height — regardless of base shape. Euclid proved it; you can also see it physically, since three suitably-shaped pyramids pack together to fill a prism with no gaps. So this pyramid's 249.42 is precisely a third of the hexagonal prism's 748.25.
The apothem trap
Volume only needs the base and the vertical height. Surface area is where it gets slippery, because the slant height is built from the apothem, and it is easy to reach for the side length instead.
| Distance | From → to | Formula | Value |
|---|---|---|---|
| Side length a | corner → adjacent corner | — | 6 |
| Apothem | centre → midpoint of a side | a√3/2 | 5.1962 |
| Circumradius | centre → corner | a | 6 |
| Slant height | apex → midpoint of a base edge | √(h² + apothem²) | 9.5394 |
| Lateral edge | apex → a base corner | √(h² + a²) | 10 |
The hexagon has a quirk that makes this worse: the circumradius equals the side length. That is true of no other regular polygon, and it means a plausible-looking "distance from centre" is sitting there ready to be mistaken for the apothem. Use a√3/2, which is about 86.6% of the side.
Substituting the side (6) for the apothem (5.196) gives a slant height of 10 rather than 9.539 — that is the lateral edge, and it inflates the lateral surface area by about 4.8%.
Surface area
Six identical triangles, each of area ½ × a × slant, so the lateral area is 6 × ½ × a × l = 3al. Add the hexagonal base if it counts.
| Component | Formula | Value | Share |
|---|---|---|---|
| Hexagonal base | (3√3/2)a² | 93.5307 | 35.3% |
| Six triangular faces | 3al | 171.7091 | 64.7% |
| Total | (3√3/2)a² + 3al | 265.2398 | 100% |
Leave the base out for anything resting on a surface — a pyramid roof, a marquee, a display piece. Include it for a closed solid such as a paperweight or a cast component.
Reference table
| Side a | Height h | Base area | Apothem | Slant l | Volume | Total SA |
|---|---|---|---|---|---|---|
| 3 | 10 | 23.3827 | 2.5981 | 10.3320 | 77.9423 | 116.3706 |
| 4 | 6 | 41.5692 | 3.4641 | 6.9282 | 83.1384 | 124.7077 |
| 5 | 12 | 64.9519 | 4.3301 | 12.7574 | 259.8076 | 256.3122 |
| 6 | 8 | 93.5307 | 5.1962 | 9.5394 | 249.4153 | 265.2398 |
The a = 4, h = 6 row is worth a second look. Its slant height (6.9282) is exactly twice its apothem (3.4641), and when that happens the lateral area comes out at exactly double the base area — 83.1384 against 41.5692. The reason is structural: the base is ½ × perimeter × apothem and the lateral area is ½ × perimeter × slant, so their ratio is simply slant ÷ apothem. Double the apothem, double the base area.
Common mistakes & pro tips
| Mistake | What happens | Fix |
|---|---|---|
| Using the side as the apothem | Slant becomes the lateral edge; lateral area +4.8% | Apothem is a√3/2 ≈ 0.866a |
| Forgetting the ⅓ | Answer 3× too big — that is the prism | Pyramids always take a third |
| Using slant height as vertical height in the volume | Over-states the volume | Volume needs the vertical height only |
| Across-corners entered as across-flats | Volume 33% too high | AC = 2a, AF = a√3 — pick the right mode |
| Multiplying the triangle area by 6 and using 3al | Double-counted | 3al already covers all six faces |
| Including the base on a roof | Over-orders material | Lateral only when it sits on something |
Pro tip. Across-flats is the easiest hexagon measurement to take accurately, and it doubles as a shortcut: the apothem is exactly half the across-flats distance. Measure AF, halve it, and you have the number the slant-height calculation actually wants — no √3 arithmetic required.
How to use this calculator
- Choose which base width you measured — side length, across flats or across corners.
- Enter that measurement, then the vertical height from the centre of the base up to the apex.
- Read the breakdown: base area, apothem, slant height, lateral edge, lateral and total surface area, and the equivalent prism volume.
- Switch the output unit for capacity in litres or gallons if the pyramid is a container or hopper.
Frequently asked questions
What is the formula for the volume of a hexagonal pyramid?
Volume is one third of the hexagonal base area times the vertical height. Since the base area is (3√3/2)a², the whole thing simplifies to V = (√3/2)a²h ≈ 0.866 a²h. A pyramid with a side of 6 and a height of 8 has a volume of 249.42 cubic units.
Why is there a one third in the pyramid formula?
Because any pyramid occupies exactly one third of the prism sharing its base and height — true for every base shape, and proved by Euclid. You can see it physically: three suitably-shaped pyramids pack together to fill a prism with no gaps. So a hexagonal pyramid of side 6 and height 8 holds 249.42 against the matching prism's 748.25.
What is the slant height of a hexagonal pyramid?
It runs from the apex down the centre of a triangular face to the midpoint of a base edge. Find it with Pythagoras using the vertical height and the apothem, not the side length: l = √(h² + apothem²). For a side of 6 the apothem is 5.196, so with a height of 8 the slant height is 9.539.
What is the difference between the apothem and the side length?
The apothem is the distance from the centre to the midpoint of a side; the side length is the length of an edge. For a regular hexagon the apothem is a√3/2 — about 86.6% of the side. Using the side in place of the apothem when finding the slant height is the most common error on this shape.
How do I find the surface area of a hexagonal pyramid?
Add the hexagonal base to the six triangular faces. The base is 2.598a² and the lateral area is 3al, since each of the six triangles is ½ × a × l. For a side of 6 and height of 8, the base is 93.53 and the lateral area 171.71, giving 265.24 square units.
How is a hexagonal pyramid different from a hexagonal prism?
A prism has two parallel hexagonal ends joined by six rectangles, so volume is base × length. A pyramid has one hexagonal base and six triangles meeting at a point, so volume is ⅓ × base × height. With the same base and height, the pyramid holds exactly one third as much.
Can I use the across-flats measurement instead of the side length?
Yes — across-flats is a√3, so divide by √3 to recover the side. Conveniently, across-flats is exactly twice the apothem, so halving it gives you the apothem directly for the slant-height step. This calculator accepts either input and converts between them.
Is the slant height the same as the lateral edge?
No. The slant height runs to the midpoint of a base edge and uses the apothem in Pythagoras. The lateral edge runs to a corner and uses the full side length, since for a regular hexagon the centre-to-corner distance equals the side. For side 6 and height 8, the slant height is 9.539 but the lateral edge is 10. Only the slant height belongs in the area formula.