Builders of roofs, spires and pyramids need the angles at which the faces and edges meet the base to cut rafters and panels. The Great Pyramid’s faces rise at about 51.8°.
How to use the Pyramid Angle Calculator
- Base
- Base side a
- Height H
Results update as you type, and everything is calculated in your browser.
How it works
Choose the base polygon and enter its side and the pyramid height. The face angle uses the apothem of the base; the edge angle uses its circumradius.
Inputs explained
| Field | What to enter |
|---|---|
| Base | Choose one: Triangle, Square, Pentagon, Hexagon … |
| Base side a | Units: cm, m, mm, in, ft, yd … |
| Height H | Units: cm, m, mm, in, ft, yd … |
What each result means
- Base apothem = [Base side a]÷(2×tan(pi÷[Base]))
- Base circumradius = [Base side a]÷(2×sin(pi÷[Base]))
- Slant height = sqrt([Height H]^2 + [Base apothem]^2)
- Lateral edge = sqrt([Height H]^2 + [Base circumradius]^2)
- α – angle between a face and the base = deg(atan([Height H]÷[Base apothem]))
- β – angle between a lateral edge and the base = deg(atan([Height H]÷[Base circumradius]))
- γ – apex angle of a face = 2×deg(atan([Base side a]÷(2×[Slant height])))
- δ – base angle of a face = (180 – [γ – apex angle of a face])÷2
Example values
How the main result changes when Base side a changes and everything else stays at the example values above:
| Base side a (cm) | α – angle between a face and the base (°) |
|---|---|
| 115 | 68.5837 |
| 172.5 | 59.5302 |
| 230 | 51.8877 |
| 287.5 | 45.5624 |
| 345 | 40.3597 |
| 460 | 32.5131 |
Formula
Apothem r = a/(2 tan(π/n)), circumradius R = a/(2 sin(π/n))
Slant s = √(H² + r²), edge e = √(H² + R²)
α = arctan(H/r), β = arctan(H/R), γ = 2·arctan(a/(2s)), δ = (180° − γ)/2
Worked example
Inputs — Base: Square; Base side a: 230 cm; Height H: 146.6 cm.
Results — Base apothem = 115 cm; Base circumradius = 162.63456 cm; Slant height = 186.323804 cm; Lateral edge = 218.955612 cm; α – angle between a face and the base = 51.887679 °; β – angle between a lateral edge and the base = 42.031734 °; γ – apex angle of a face = 63.366192 °; δ – base angle of a face = 58.316904 °.
Key terms
- Dihedral angle
- The angle between two planes, such as a face and the base.
Frequently asked questions
What is the face angle of the Great Pyramid?
About 51.8°.
Why is the edge angle smaller than the face angle?
The edge runs to a corner, which is farther from the centre than the middle of a side.
What happens as the height grows?
All angles at the base approach 90°.
Do these formulas work for oblique pyramids?
No – only for right regular pyramids.