How this calculator works
A right square pyramid has its apex directly above the centre of a square base. Three angles describe how steeply its faces and edges rise. The dihedral angle is measured between the flat base and a triangular face, using the slant height and half the base side. The lateral edge angle measures a corner edge above the base, with the distance from the base centre to a corner as the horizontal run. The apex angle is the angle between the two equal edges of one triangular face at the top of the pyramid.
Formula
Dihedral angle = arctan(2h/a); lateral edge angle = arctan(√2 h/a); apex face angle = 2 arcsin(a / (2 × lateral edge)); slant height = √(h² + (a/2)²).
Worked example
For a base side of 4 m and a height of 3 m, the dihedral angle is arctan(1.5) ≈ 56.310°, the lateral edge rises at about 46.686°, and the apex angle of each triangular face is about 58.034°. The slant height is √13 ≈ 3.606 m and the lateral edge is √17 ≈ 4.123 m.
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Frequently asked questions
What is the dihedral angle?
It is the angle between two flat surfaces that meet along a line. Here it is the angle between the horizontal base and one of the sloping triangular faces, measured inside the pyramid.
Is the slant height the same as the lateral edge?
No. The slant height runs up the middle of a triangular face from the midpoint of a base edge. The lateral edge runs up a corner from a base vertex. The lateral edge is longer because the corner is farther from the base centre.
Does this work for a pyramid with a rectangular base?
No. This calculator assumes a square base so that the face angles are consistent. A rectangular base produces two different dihedral angles, one for each pair of opposite faces.