Volume of a Pyramid
Geometry & MeasurementThe volume of a pyramid is one-third times the base area times the height.
Formula
V = \frac{1}{3} \times B \times h
Definition
The volume of a pyramid is one-third of the base area times the height: $V = (1/3) \times B \times h$. The base area $B$ depends on the shape of the base (square, triangle, etc.).
Example
A square pyramid with base $6$ cm $\times$ $6$ cm and height $4$ cm: $B = 36$ cm$^2$, $V = (1/3)(36)(4) = 48$ cm$^3$. A triangular pyramid with triangular base area $10$ cm$^2$ and height $9$ cm: $V = (1/3)(10)(9) = 30$ cm$^3$.
Key Insight
Three identical square pyramids can be assembled to form a cube. So each pyramid holds $1/3$ of the cube's volume. This is where the $1/3$ comes from: a pyramid always holds exactly $1/3$ of the prism with the same base and height.
Definition
For any pyramid with base area $B$ and perpendicular height $h$, $V = (1/3)Bh$. The formula holds regardless of the base polygon shape (triangle, square, hexagon, etc.) and for oblique pyramids, where $h$ is still the perpendicular distance from apex to base plane.
Example
The Great Pyramid of Giza: original base $230.4$ m $\times$ $230.4$ m ($B = 53{,}084$ m$^2$), height $146.5$ m. $V = (1/3)(53084)(146.5) = 2.59 \times 10^6$ m$^3 = 2.59$ million cubic meters.
Key Insight
Any prism can be cut into exactly $3$ congruent pyramids. This tangible dissection proof is the most satisfying explanation for the $1/3$ factor, and it works with any prism, not just rectangular ones.
Definition
For a pyramid with base area $B$ and height $h$, $$V = \int_0^h B(1-z/h)^2 \, dz = Bh\int_0^1 (1-t)^2 \, dt = \frac{Bh}{3}.$$ The cross-section at height $z$ is a scaled copy of the base with linear scale factor $(1-z/h)$, giving area $B(1-z/h)^2$. Integration of a quadratic function yields the $1/3$ factor.
Example
A frustum (pyramid with top cut off) with base area $B_1$, top area $B_2$, height $h$: $V = (h/3)(B_1 + B_2 + \sqrt{B_1 B_2})$. This formula, derived in ancient Egypt (Moscow Mathematical Papyrus, ca. 1850 BCE), is one of the earliest known volume calculations.
Key Insight
The Moscow Mathematical Papyrus gives the correct frustum formula without derivation, suggesting ancient Egyptians had an intuitive or empirical method. The formula was not rigorously proved until the era of calculus, $3500$ years later, testifying to the remarkable mathematical sophistication of ancient Egypt.