Pyramid Angle Calculator

A pyramid has several angles: the slant angle (how steep the faces tilt from horizontal), the apex angle (the angle at the top point), the base angle (angle between the face and base edge), and dihedral angles (angles between adjacent faces).

The slant angle θ = arctan(h/(a/2)), where h is the vertical height and a is the base side length. This gives the angle the triangular face makes with the horizontal base plane.

The apex angle is the angle at the top of the pyramid. The apex half-angle = arctan((a/2)/h), and the full apex angle = 2 × arctan((a/2)/h). A taller pyramid has a smaller (sharper) apex angle.

The dihedral angle is the angle between two adjacent triangular faces. For a regular square pyramid, it can be calculated using the formula cos(θ) = (a²/4) / (a²/4 + h²), measured at the edge where two faces meet.

A taller pyramid (larger h/a ratio) has a steeper slant angle. When h = a/2, the slant angle is exactly 45°. When h > a/2, the angle exceeds 45°. When h < a/2, the angle is less than 45°.

The slant height l is the distance from the apex to the midpoint of a base edge, measured along the triangular face. It equals √(h² + (a/2)²), where h is the vertical height and a is the base side.

This calculator assumes a square base. For rectangular or polygonal bases, the angle calculations become more complex as the slant angles differ for each face. You would need to calculate each face angle separately.

Evidence suggests they used simple tools like set squares, plumb bobs, and measuring ropes. The seked system (ratio of horizontal run to vertical rise) allowed them to specify and maintain consistent slopes during construction.

Steeper pyramids (higher slant angles) distribute weight more vertically, reducing outward thrust. The optimal angle for stability in dry-stone construction is approximately 51.8°, which happens to match the Great Pyramid.

The face area = ½ × a × l, where l is the slant height. Since l = (a/2)/cos(θ), a steeper angle (larger θ) increases the slant height and therefore the face area, even though the pyramid height stays the same.

The Great Pyramid has a slant angle of approximately 51°50' (51.83°). This creates a near-perfect proportions where the ratio of half-perimeter to height equals π, though this may be coincidental.

A golden pyramid has a slant angle where the cross-section forms a golden ratio relationship: the ratio of slant height to half-base equals φ (≈ 1.618). This gives a slant angle ≈ 58.28°.

For a regular square pyramid, the dihedral angle θ_d satisfies cos(θ_d) = cos(θ_s) × sin(θ_s) / sin(α), where θ_s is the slant angle and α is related to the base geometry. Alternatively, use the dot product of face normals.

No. The apex angle is always between 0° and 180°. An apex angle of 180° would mean the pyramid is completely flat (zero height). As height increases, the apex angle decreases toward 0°.

Use the triangle formed by the height, half-base, and slant height. Verify: slant height² = h² + (a/2)² (Pythagorean theorem). Also verify: slant angle + apex half-angle = 90° (complementary angles in the right triangle).

The angle between the triangular face and the base edge at the bottom is the same as the slant angle θ = arctan(h/(a/2)). This is because the face rises from the base edge at this angle.

A cone's slant angle works the same way: θ = arctan(h/r), where r is the base radius. The cone is the limit of a pyramid as the number of base sides approaches infinity.

The solid angle Ω at the apex of a square pyramid is: Ω = 4 × arcsin(sin²(α)), where α is related to the apex half-angle. This measures the "amount of sky" visible from the apex looking outward.

The calculator uses IEEE 754 double-precision floating-point arithmetic with approximately 15 significant digits. For any practical application in architecture, engineering, or education, this precision is more than sufficient.

Yes. Rearranging: h = (a/2) × tan(θ). If you know the base side and desired slant angle, multiply the half-base by the tangent of the angle to get the required height.

The Pyramid Angle Calculator uses standard mathematical formulas validated against authoritative references. The specific formula is displayed in the calculator interface with a detailed explanation of each variable.

Each calculator includes a step-by-step worked example showing exactly how the formula is applied. You can follow these steps with pen and paper to verify any result.

The Pyramid Angle Calculator accepts numeric inputs including integers and decimals. Invalid inputs (letters, special characters) are rejected with clear error messages.