Pyramid Angle Calculator

What is the Pyramid Angle Calculator?

A regular pyramid consists of a regular polygon base and congruent triangular lateral faces meeting at a single apex above the base center. Because of the three-dimensional geometry, pyramids have several distinct angles: the slope angle between a triangular face and the base, the corner angle between a lateral edge and the base, the apex angle of each triangular face, and the dihedral angle between adjacent lateral faces. If you also need surface area and volume calculations, check our pyramid calculator or solve right triangles with our pythagorean calculator.

The Key Angles of a Regular Pyramid

For a regular pyramid whose base is a regular $n$-sided polygon with edge length $a$ and vertical height $h$, the key angles are defined as follows:

1. Face Slope Angle ($\alpha$)

This is the angle between any lateral triangular face and the base plane. It is measured inside the right triangle formed by the vertical height $h$ and the base apothem (inradius $r_{in}$):

$$r_{in} = \frac{a}{2 \tan(\pi / n)}, \quad \tan(\alpha) = \frac{h}{r_{in}} \implies \alpha = \arctan\left(\frac{h}{r_{in}}\right)$$

2. Edge Corner Angle ($\beta$)

This is the angle between a lateral edge (connecting the apex to a base vertex) and the base plane. It is measured inside the right triangle formed by height $h$ and the base circumradius $r_{out}$:

$$r_{out} = \frac{a}{2 \sin(\pi / n)}, \quad \tan(\beta) = \frac{h}{r_{out}} \implies \beta = \arctan\left(\frac{h}{r_{out}}\right)$$

3. Apex Angle of the Face ($\gamma$)

Each lateral face is an isosceles triangle with base $a$ and legs equal to the lateral edge length $e = \sqrt{h^2 + r_{out}^2}$. The apex angle at the top vertex is:

$$\sin\left(\frac{\gamma}{2}\right) = \frac{a}{2e} \implies \gamma = 2 \arcsin\left(\frac{a}{2e}\right)$$

4. Dihedral Angle Between Adjacent Lateral Faces ($\delta$)

The dihedral angle is the interior angle between two adjacent triangular faces sharing a common lateral edge. Using the face normal vectors tilted by angle $\alpha$ and rotated by central angle $\phi = 2\pi/n$:

$$\cos(\pi - \delta) = \sin^2(\alpha) \cos\left(\frac{2\pi}{n}\right) + \cos^2(\alpha)$$

Example: The Great Pyramid of Giza

The Great Pyramid of Giza has a square base ($n = 4$) with original base side length approximately $a = 230.4\text{ m}$ and original height $h = 146.7\text{ m}$:

Base inradius (apothem): $r_{in} = 230.4 / 2 = 115.2\text{ m}$
Face slope angle: $\alpha = \arctan(146.7 / 115.2) \approx 51.86^\circ$
Base circumradius: $r_{out} = 115.2 \times \sqrt{2} \approx 162.92\text{ m}$
Corner edge angle: $\beta = \arctan(146.7 / 162.92) \approx 42.00^\circ$
Dihedral angle between faces: $\delta \approx 112.42^\circ$

Frequently Asked Questions

What is the difference between face slope angle and edge corner angle?

The face slope angle ($\alpha$) measures the incline of the flat triangular face relative to the base, measured along the slant height to the midpoint of a base side. The edge corner angle ($\beta$) measures the steeper incline of the corner rib line from the apex down to a base vertex. Because vertices are farther from the center than side midpoints, $\beta$ is always smaller than $\alpha$.

What are the angles of the Great Pyramid of Giza?

The Great Pyramid of Giza has a face slope angle of approximately $51.86^\circ$ and a corner edge angle of approximately $42.00^\circ$. Its apex face angle is about $63.40^\circ$, and the dihedral angle between adjacent faces is about $112.42^\circ$.

How do you calculate angles for a hexagonal pyramid?

For a regular hexagon base ($n = 6$), the apothem is $r_{in} = a \sqrt{3} / 2$ and the circumradius is $r_{out} = a$. The face slope angle is $\arctan(h / (a\sqrt{3}/2))$ and the edge angle is $\arctan(h / a)$.

What is the dihedral angle of a pyramid?

The dihedral angle is the interior angle between two adjacent lateral triangular faces that meet at a shared edge. It is useful in woodworking, masonry, and carpentry when cutting bevel joints for compound roofs or pyramid structures.

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