Pyramid Angle Calculator
Enter the base side length, height, and number of base sides to find all four standard angles of a regular right pyramid: the face-inclination angle (alpha), the lateral-edge angle (beta), the triangular-face base angle (gamma), and the apex angle (delta). The calculator also gives you the slant height, lateral edge length, and base apothem. Results update instantly as you type.
The four angles of a regular right pyramid
A regular right pyramid has a regular polygon as its base and an apex directly above the base center. Every triangular face is an isosceles triangle and is identical to every other face. Four distinct angles fully describe the geometry: - Alpha (face-inclination angle): the angle between the slant height (the line from apex to the midpoint of a base edge) and the base plane. This is the most commonly quoted angle, defining how steep the faces are. - Beta (edge-inclination angle): the angle between a lateral edge (apex to base corner) and the base. Always smaller than alpha because corners are farther from the center than midpoints. - Gamma (face base angle): the angle at the base corner inside one triangular face. Determined by how wide the face is relative to its height. - Delta (apex angle): the angle at the very tip of each triangular face. Together with two gamma values, it totals 180 deg.
Formulas used in this calculator
All calculations use exact trigonometry for a regular n-sided right pyramid with base side a and height h. Base apothem (CM): the perpendicular distance from the base center to the midpoint of one base edge. CM = a / (2 * tan(pi / n)) Circumradius (AC): distance from base center to a base corner. AC = a / (2 * sin(pi / n)) Slant height (MO): distance from apex to midpoint of a base edge. MO = sqrt(h^2 + CM^2) Lateral edge (BO): distance from apex to a base corner. BO = sqrt(h^2 + AC^2) Alpha: tan(alpha) = h / CM, so alpha = atan(h / CM) Beta: tan(beta) = h / AC, so beta = atan(h / AC) Gamma: cos(gamma) = CM / MO, so gamma = acos(CM / MO) Delta: delta = 180 - 2 * gamma For a square base (n = 4): CM = a / 2, so alpha = atan(2h / a). That is why the Great Pyramid of Giza, with h approximately equal to a/2 * phi, has alpha close to 51.84 deg.
How to use the calculator and what the results mean
Select the number of base sides (3 for a tetrahedron-like shape, 4 for the classic Egyptian square pyramid, 6 for a hexagonal pyramid, etc.), then enter the base side length and the perpendicular height. All four angles and three key distances update instantly. If you want to design a pyramid with a specific face-inclination angle, first pick your base size and then solve for the height: h = CM * tan(alpha). For a square pyramid that means h = (a / 2) * tan(alpha). Unit toggle: switch between metric (metres) and imperial (feet). All lengths convert automatically; angles are always in degrees. Real-world uses: woodworking and carpentry (cutting pyramid roof panels), architecture and landscaping, 3-D printing and game design, and archaeology (reproducing or analyzing historical pyramid proportions).
Notable pyramid proportions
The Great Pyramid of Khufu at Giza has a face-inclination angle of about 51.84 deg - very close to the angle produced when the height equals half the base times the golden ratio phi (approximately 1.618). The exact relationship is alpha = atan(4h / (2a)) = atan(phi) for the phi pyramid, giving 51.83 deg. Egypt's Red Pyramid uses a shallower angle of 43.36 deg, giving it a noticeably wider, lower profile. The Bent Pyramid starts at 54.46 deg at the base and flattens to 43.36 deg partway up, possibly because of structural problems during construction. An equilateral pyramid (every triangular face is an equilateral triangle) has alpha = 60 deg regardless of scale. A regular tetrahedron is the special case where the base is also an equilateral triangle.
Famous pyramid face-inclination angles (alpha)
| Pyramid / type | Alpha (deg) | Notes |
|---|---|---|
| Flat / low-slope | < 30 | Very wide base, shallow faces |
| Egyptian Red Pyramid (Dahshur) | 43.36 | Oldest true smooth-sided pyramid |
| Egyptian Bent Pyramid (upper) | 43.36 | Lower section is 54.46 deg |
| Great Pyramid of Giza (Khufu) | 51.84 | Most iconic, phi-ratio approximation |
| Golden ratio pyramid | 51.83 | Height = base/2 * (1/phi), same as Khufu |
| Equilateral-face pyramid | 60.00 | Each triangular face is equilateral |
| Steep / obelisk-like | > 75 | Tall, narrow pyramid |
The alpha angle (face inclination to the base) defines the visual steepness of a pyramid. These historic and geometric reference values are all square pyramids (n = 4).
Frequently asked questions
What is the difference between alpha and beta in a pyramid?
Alpha is the angle between the slant height (from apex to base-edge midpoint) and the base. Beta is the angle between the lateral edge (from apex to base corner) and the base. Because base corners are farther from the center than base-edge midpoints, the lateral edge is longer than the slant height for the same vertical rise, so beta is always smaller than alpha for any regular right pyramid.
How do I find the angle of a pyramid if I only know the slant height and base side?
For a square pyramid, alpha = atan(slant height / apothem), where apothem = a / 2 (half the base side). First use the slant height to recover the height: h = sqrt(slant^2 - (a/2)^2), then feed h and a into this calculator. For other polygon bases, the apothem is a / (2 * tan(pi / n)).
Why does the Great Pyramid have an angle close to 51.84 degrees?
The most widely accepted scholarly explanation is that the builders used a seked ratio of 5.5 palms per cubit of rise, which produces a face-inclination angle of atan(28/22) = 51.84 deg. A popular modern observation is that this is nearly identical to atan(phi) where phi is the golden ratio (1.618), giving 51.83 deg. Whether or not the Egyptians intended the golden-ratio relationship is debated, but the seked explanation is well-documented.
What is the face angle of an equilateral pyramid?
An equilateral pyramid is one in which every triangular face is an equilateral triangle. That means each face angle (gamma) is 60 deg, the apex angle (delta) is also 60 deg, and the face-inclination angle (alpha) is arctan(sqrt(2)) which is about 54.74 deg for a triangular base. For a square base, an equilateral-face condition sets alpha = 60 deg.
Can I use this calculator for roof pitch or hip roof angles?
Yes. A hip roof over a square or rectangular footprint is geometrically a pyramid. Enter the half-width of the building as the base apothem and the ridge height to find the roof pitch angle (alpha). For a rectangular roof the two pairs of sides have different apothems and therefore different alpha angles; use the calculator twice, once for each pair.
How does the number of base sides affect the angles?
Increasing the number of sides (at constant height and side length) brings the base apothem (CM) closer to the circumradius (AC), so alpha and beta converge toward each other. For a very large number of sides the pyramid approaches a cone and the two angles become equal. Gamma and delta also change: more sides means each face is narrower, increasing gamma and decreasing delta.