Pyramid Angle Calculator — Square Pyramid Angles & Dimensions
Find every important angle of a right square pyramid — face slope, lateral edge inclination, apex angle, and the dihedral angle between adjacent faces — from the base side and height.
Angle between each lateral face and the base
- 1
Apothem (half the base side)
6 ÷ 2 = 3 - 2
Slant height l = √(h² + apothem²)
√(8² + 3²) = 8.544Distance from the apex to the midpoint of a base edge, along the face surface. - 3
Face inclination angle = arctan(h ÷ apothem)
arctan(8 ÷ 3) = 69.44
How does this calculator work?
For a right square pyramid with base side b and height h: slant height l = √(h² + (b/2)²); face inclination = arctan(2h/b); lateral edge angle = arctan(h√2/b). The face apex angle = 2 × arctan(b/(2l)), and the dihedral angle between adjacent faces is derived from the face normals.
Formula
How this is calculated
A right square pyramid has a square base of side b and an apex directly above the centre at height h. Five distinct angles describe how its faces and edges lean.
The slant height l = √(h² + (b/2)²) is the distance from the apex to the midpoint of any base edge, measured along the surface. The face inclination angle — the angle each triangular face makes with the base — equals arctan(h / (b/2)), because the slant height is the hypotenuse of a right triangle whose legs are h and the base apothem b/2. A steeper, narrower pyramid gives a larger face angle.
Each lateral edge (ridge) connects the apex to a corner of the square base. The horizontal distance from the base centre to a corner is b/√2, so the lateral edge makes an angle arctan(h / (b/√2)) with the base — always greater than the face angle. The apex angle of each triangular face is 2 × arctan((b/2) / l), and the base angles of each face are each (180° − apex angle) / 2. The dihedral angle between two adjacent lateral faces (measured along their shared lateral edge) is computed from the dot product of the outward face normals and is typically an obtuse angle for reasonable proportions.
Frequently asked questions
The slant height is the distance from the apex to the midpoint of any base edge, measured along the surface — not the lateral edge to a corner. For a square pyramid with base b and height h, slant height l = √(h² + (b/2)²).
The dihedral angle between two adjacent lateral faces is the angle you would measure if you placed a protractor along the shared lateral edge (ridge). For a square pyramid it is computed from the dot product of the two face normal vectors.
The face inclination angle measures how steeply a triangular face rises from the base, using the slant height path. The lateral edge angle measures the steeper rise along the ridge (corner edge) to the apex. The edge angle is always larger than the face angle for any square pyramid.
Also known as
TG we-Calculate Editorial Team. (2026). Pyramid Angle Calculator — Square Pyramid Angles & Dimensions [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/pyramid-angle-calculator
TG we-Calculate Editorial Team. "Pyramid Angle Calculator — Square Pyramid Angles & Dimensions." TG we-Calculate. 2026. https://we-calculate.com/calculator/pyramid-angle-calculator.
TG we-Calculate Editorial Team, "Pyramid Angle Calculator — Square Pyramid Angles & Dimensions," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/pyramid-angle-calculator
@misc{wecalculate_pyramid_angle_calculator, title = {Pyramid Angle Calculator — Square Pyramid Angles & Dimensions}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/pyramid-angle-calculator}}, year = {2026}, note = {TG we-Calculate} }
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