Use our pyramid angle calculator to find all the possible angles in a pyramid — we'll ask you a few parameters, and do all the math for you! Here you will learn:
- The types of angles in a pyramid;
- How to calculate the angles of a pyramid;
- An example of how to calculate the square pyramid angles; and
- How to use our pyramid angle calculator.
What is a pyramid?
A pyramid is a solid figure with a polygonal base. A triangular face corresponds to each side of the base, joining at the apex.
The base can be of any shape; however, regular pyramids are easier to study. The base of a regular pyramid is a regular polygon. We can find, for example:
- Triangular pyramids, with an equilateral triangle as the base;
- Square pyramids, where the base is, of course, a square;
- Hexagonal pyramids; and
- Well, infinitely more.
If the apex lies above the centroid of the base (the geometric mean point of the polygon), we call the pyramid a right pyramid; otherwise, we have an oblique pyramid. Here we will analyze only right pyramids.
The angles of a pyramid
There are a lot of angles in a pyramid, but luckily in a regular pyramid, most of them are identical. Let's identify them!
We can identify the angle between the faces' vertical medians (the slant height) and the base. It defines how "slender" a pyramid is. We call this angle α.
The angle between the edge and the base, β, is always smaller than α in pyramids with a convex base: the corner of the base is farther from the centroid than the center of its sides.
We can identify other angles, also related to the pyramid's height. They lie on each face, and since in a regular pyramid a face is an isosceles triangle, the two angles at the bottom are identical. We identify them with the letter γ. The angle on the top, at the apex, gets smaller for tall and slender pyramids. We call it δ.
α — the angle between a face and the base (how slender the pyramid is);
β — the angle between a lateral edge and the base;
γ — the two base angles of a triangular face; and
δ — the apex (tip) angle of a triangular face.
How to calculate the pyramid angles
Take a regular pyramid whose base is a regular polygon with n sides, each of length a, and whose height is h. Two base measurements do all the work:
- The apothem of the base (center to the midpoint of a side): m = a / (2·tan(180°/n));
- The circumradius of the base (center to a vertex): R = a / (2·sin(180°/n)).
The slant height (the median of a face) and the lateral edge follow from the Pythagorean theorem:
Then the four angles are:
β = arctan(h / R)
γ = arctan(l / (a/2))
δ = 180° − 2γ
How to calculate the angles of a square pyramid
For a square base (n = 4) the two base measurements become simple: the apothem is m = a/2 and the circumradius is R = a/√2.
Let's take a square pyramid with base edge a = 4 cm and height h = 3 cm:
l = √(3² + 2²) = √13 ≈ 3.61 cm
α = arctan(3 / 2) ≈ 56.31°
β = arctan(3 / 2.83) ≈ 46.69°
γ = arctan(3.61 / 2) ≈ 61.00°
δ = 180° − 2·61.00° ≈ 58.00°
As expected, β < α: the lateral edge reaches a corner of the base, which sits farther from the center than the midpoint of a side.
How to use our pyramid angle calculator
- Pick the base shape — the number of sides n (triangular, square, pentagonal, hexagonal, and so on).
- Choose your unit system — metric (cm) or US/Imperial (in).
- Enter the base edge length a of the polygon.
- Enter the pyramid height h.
The calculator instantly returns all four angles α, β, γ, and δ, together with the base apothem, circumradius, slant height, and lateral edge.
Unit Systems
Metric System
Enter the base edge and height in centimeters (cm). The lengths in the results are shown in the same unit; the angles are always in degrees.
US / Imperial System
Enter the base edge and height in inches (in). The angles do not depend on the unit — only on the ratio of height to base — so they stay the same whichever system you pick.
FAQs
What angles does a pyramid have?
A regular pyramid has four characteristic angles: α between a face and the base, β between a lateral edge and the base, γ at the base of each triangular face, and δ at the apex (tip) of each face.
Why is β smaller than α?
Because a lateral edge reaches a base corner, while the face median reaches the midpoint of a base side. In a convex base the corner is farther from the centroid, so the edge is more inclined toward the base and its angle is smaller.
Do the angles change with the unit system?
No. Angles depend only on the ratio between the height and the base dimensions, not on the absolute lengths, so a pyramid measured in centimeters or inches has exactly the same angles.
What are the angles of a regular tetrahedron?
A regular tetrahedron is a triangular pyramid whose faces are all equilateral triangles. Its face–base angle α is about 70.53°, and every face angle (γ and δ) equals 60°.