Free Pyramid Volume Calculator

Pyramid Volume Formula

Any pyramid’s volume can be found by multiplying its base area by its perpendicular height and then taking one‑third of that product. This universal relation holds for every pyramid, regardless of whether the base is a triangle, square, pentagon, or any other polygon, and works for both right and oblique pyramids:

$$V = \frac{1}{3} B h$$

where $B$ is the area of the base and $h$ is the height measured from the apex to the base plane.

When the base is a regular polygon (all sides equal, all interior angles equal), the base area can be expressed using only the side length $s$ and the number of sides $n$. Substituting that expression into the general formula gives a compact form that covers all regular pyramids:

$$V = \frac{n}{12} s^{2} h \cot\left(\frac{\pi}{n}\right)$$

The following sections apply this general rule to the most common pyramid shapes, providing ready‑to‑use formulas for each.

Volume of a Square Pyramid (n = 4)

For a square base, $\cot(\pi/4) = 1$, so the volume simplifies to the well‑known relation:

$$V_{\text{square}} = \frac{1}{3} s^{2} h$$

A classic example is the Great Pyramid of Giza (Cheops). Its base side length is about 230.3 m. Originally it stood 146.6 m high, giving an original volume of roughly 2,591,795 m³. Erosion has reduced its height to 138.5 m, lowering the present‑day volume to approximately 2,448,592 m³.

Volume of a Triangular Pyramid (Tetrahedron, n = 3)

When the base is an equilateral triangle, $\cot(\pi/3) = 1/\sqrt{3}$, and the volume becomes:

$$V_{\text{triangular}} = \frac{\sqrt{3}}{12} s^{2} h$$

A special case is the regular tetrahedron, where all four faces are equilateral triangles. Its height is linked to the edge length $a$ by $h = a\sqrt{3}/6$. Inserting this relation yields a formula that depends only on the edge:

$$V_{\text{tetrahedron}} = \frac{a^{3}}{6\sqrt{2}}$$

For instance, a tea‑pyramid sachet with a height of 1.2 inches and a base side of 1.5 inches has a volume of about 0.39 cubic inches using the first triangular formula.

Volume of a Hexagonal Pyramid (n = 6)

A regular hexagonal pyramid has a base with six equal sides. Since $\cot(\pi/6) = \sqrt{3}$, the volume expression condenses to:

$$V_{\text{hexagonal}} = \frac{\sqrt{3}}{2} s^{2} h$$

This formula is directly usable for any regular hexagon‑based pyramid.

Volume of a Pentagonal Pyramid (n = 5)

For a regular pentagon base, $\cot(\pi/5) = \sqrt{\dfrac{25+10\sqrt{5}}{5}}$. Substituting into the general equation gives:

$$V_{\text{pentagonal}} = \frac{\sqrt{25+10\sqrt{5}}}{12} s^{2} h$$

Volume of an Octagonal Pyramid (n = 8)

With $\cot(\pi/8) = 1+\sqrt{2}$, the octagonal pyramid volume formula becomes:

$$V_{\text{octagonal}} = \frac{2(1+\sqrt{2})}{3} s^{2} h$$

These shape‑specific forms allow rapid volume estimation without needing to compute the base area separately.

Pyramid Geometry & Naming

A pyramid with an n‑sided base always has $n+1$ faces (n triangular side faces plus the base), $2n$ edges, and $n+1$ vertices. The naming follows directly from the base’s shape, as summarised in the table below.

Base Shape	Faces	Edges	Vertices	Pyramid Name
Triangle	4	6	4	Triangular pyramid (tetrahedron)
Square	5	8	5	Square pyramid
Pentagon	6	10	6	Pentagonal pyramid
Hexagon	7	12	7	Hexagonal pyramid
Heptagon	8	14	8	Heptagonal pyramid
Octagon	9	16	9	Octagonal pyramid

Using the calculator is straightforward: you may enter the base area and height directly, or specify the base shape, side length, and height when dealing with a regular pyramid. The tool then applies the appropriate formula and returns the volume instantly.

This guide covers the essential principles behind pyramid volume calculation – from the fundamental formula through the specific equations for square, triangular, hexagonal, pentagonal, and octagonal pyramids – enabling you to compute the volume of virtually any regular pyramid.