Last updated: 2026-05-18

What a Pyramid Is

A pyramid is a solid with a polygonal base and triangular faces that meet at a single point — the apex. A right pyramid has its apex directly above the center of the base; a rectangular pyramid has a rectangular base, with the square pyramid as a special case. The volume, base area, slant height, and surface areas all follow from the base dimensions and the vertical height.

Formulas

With base length $l$, base width $w$, and vertical height $h$:

Base area:

Volume — always one-third of the equivalent prism:

Slant height — the distance from the apex to the midpoint of a base edge. For a rectangular pyramid, each pair of opposite faces has its own slant height:

For a square pyramid ($l = w = a$), both slant heights are equal: $l_s = \sqrt{h^2 + (a/2)^2}$.

Lateral surface area — the combined area of the four triangular faces:

Total surface area:

The one-third factor

Any pyramid has exactly one-third the volume of a prism with the same base and height. The classical proof places three congruent square pyramids together to form a cube — each pyramid occupies exactly $\frac{1}{3}$ of the cube's volume. A more general proof uses Cavalieri's principle: at every height $z$, the cross-sectional area of the pyramid is $A_b \times (1 - z/h)^2$, which integrates to $\frac{1}{3} A_b h$ from $z = 0$ to $z = h$.

Slant height vs. vertical height

The vertical height $h$ is the perpendicular distance from the apex straight down to the base plane. The slant height $l_s$ is the distance from the apex to the midpoint of a base edge — measured along the surface of a triangular face. These are related by a right triangle: $l_s = \sqrt{h^2 + (r)^2}$ where $r$ is the distance from the center of the base to the midpoint of the corresponding edge ($r = w/2$ for the front/back faces).

In physical terms, the slant height is the measurement taken by laying a ruler flat on a triangular face, from the apex down to the midpoint of the base edge.

Worked example

The Great Pyramid of Giza has a square base of approximately 230.4 m and an original height of 146.5 m.

Base area: $A_b = 230.4^2 = 53{,}084 \text{ m}^2 \approx 5.3 \text{ ha}$

Volume: $V = \frac{1}{3} \times 53{,}084 \times 146.5 \approx 2{,}592{,}000 \text{ m}^3$

That is roughly 2.6 million cubic metres — equivalent to filling about 1,000 Olympic swimming pools.

Slant height: $l_s = \sqrt{146.5^2 + 115.2^2} = \sqrt{21{,}462 + 13{,}271} = \sqrt{34{,}733} \approx 186.4 \text{ m}$

Lateral surface area: $A_{lat} = 4 \times \frac{1}{2} \times 230.4 \times 186.4 \approx 85{,}944 \text{ m}^2 \approx 8.6 \text{ ha}$