Definition

Volume measures how much space is inside a three-dimensional object, expressed in cubic units like cubic centimeters ($\text{cm}^3$) or cubic feet ($\text{ft}^3$); each solid has a formula derived from a cross-sectional area integrated along a depth. For prisms and cylinders, $V = \text{base area} \times \text{height}$; for pyramids and cones, $V = (1/3) \times \text{base area} \times \text{height}$. Formally, volume in $\mathbb{R}^3$ is the three-dimensional Lebesgue measure, computed as a triple integral $V = \iiint_R dV$; by Cavalieri's principle, if two solids have equal cross-sectional areas at every height, they have equal volume, and for smooth solids bounded by surface $S$, the divergence theorem gives $V = (1/3)\left|\oint_S r \cdot n \, dA\right|$.