How to Use Pyramid Angle Calculator
The Pyramid Angle Calculator computes the key angles of a right square pyramid from just two inputs: the height and the base side length. Enter both values, and the calculator instantly displays the face angle, lateral edge angle, and apex dihedral angle in both degrees and radians.
- Height (h) — Enter the perpendicular height from the center of the base to the apex.
- Base Side Length (a) — Enter the length of one side of the square base.
- Review Results — The Pyramid Angle Calculator shows all three angles along with the substituted formulas so you can verify each step.
Both inputs must be positive numbers. The calculator works with any consistent unit (meters, feet, centimeters, etc.) since the angles are dimensionless ratios.
Formula & Theory - Pyramid Angle Calculator
The Pyramid Angle Calculator uses this core formula or rule: trigonometry based on the geometry of a right square pyramid:
Face Angle: θ = arctan(h / (a/2))
Lateral Edge: φ = arctan(h / (a/√2))
Apex Dihedral: α = 2 × arctan((a/2) / h)| Symbol | Meaning |
|---|---|
| h | Perpendicular height from base to apex |
| a | Side length of the square base |
| θ | Angle between triangular face and base |
| φ | Angle between lateral edge and base |
| α | Dihedral angle at the apex between opposite faces |
The face angle θ uses the apothem of the base (a/2) as the horizontal leg and h as the vertical leg of a right triangle formed by slicing the pyramid through the midpoint of a base edge. The edge angle φ uses the half-diagonal of the base (a/√2) instead, since the lateral edge runs from a corner of the square to the apex.
Assumptions and Limits
This calculator assumes a right square pyramid — the apex is directly above the center of the square base and all four triangular faces are congruent isosceles triangles. Inputs must be positive. If h or a is zero, angles are undefined.
Use Cases for Pyramid Angle Calculator
The Pyramid Angle Calculator is useful in geometry, architecture, and 3D design whenever you need to work with pyramid structures:
- Geometry learning — Verify textbook problems involving pyramid cross-sections and slant heights.
- Architectural design — Calculate roof pitch or facade angles for pyramid-shaped structures.
- 3D modeling — Set precise face normals and edge orientations in software like Blender or CAD tools.
- Scale model construction — Cut materials at exact angles for physical model pyramids.
- Pyramid structure analysis — Study the proportions of ancient Egyptian pyramids or modern pyramid buildings.
The angles output by the Pyramid Angle Calculator are fully self-consistent: if you verify them with the law of cosines or by constructing the triangle on paper, they will match exactly.