Volume
Geometry & MeasurementVolume is the measure of the three-dimensional space enclosed within a solid figure, expressed in cubic units.
Formula
\text{Varies by solid; measured in cubic units}
Definition
Volume measures how much space is inside a three-dimensional object. It is measured in cubic units, like cubic centimeters ($\text{cm}^3$) or cubic feet ($\text{ft}^3$).
Example
A box $4$ cm long, $3$ cm wide, and $2$ cm tall has volume $4 \times 3 \times 2 = 24$ cm$^3$. You could pack exactly $24$ tiny cubes (each $1$ cm $\times$ $1$ cm $\times$ $1$ cm) into that box.
Key Insight
Volume answers "how much fits inside?" while surface area answers "how much wrapping is needed?" A big box and a small box can look similar but hold very different amounts.
Definition
Volume is the three-dimensional measure of the interior of a solid, expressed in cubic units. Each solid has a formula derived from cross-sectional area integrated along a depth. For prisms and cylinders, $V = \text{base area} \times \text{height}$; for pyramids and cones, $V = (1/3) \times \text{base area} \times \text{height}$.
Example
A rectangular prism $5$ m $\times$ $4$ m $\times$ $3$ m: $V = 60$ m$^3$. A cylinder with radius $3$ cm and height $10$ cm: $V = \pi \times 9 \times 10 = 90\pi = 282.7$ cm$^3$. Doubling every dimension multiplies volume by $8$.
Key Insight
Volume scales as the cube of linear dimensions. A shape scaled by factor $k$ has volume $k^3$ times the original. This is why ants can carry objects many times their weight but giants in stories would collapse under their own mass.
Definition
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. For smooth solids bounded by surface $S$, the divergence theorem gives $V = (1/3)\left|\oint_S r \cdot n \, dA\right|$.
Example
Cavalieri's principle proves that a sphere of radius $r$ and a cylinder of radius $r$ and height $2r$ with a double cone removed have equal volumes. This elegant "method of indivisibles" precedes calculus and was used by Archimedes.
Key Insight
The factor of $1/3$ in pyramid and cone volume formulas arises from integrating a linearly decreasing cross-section. Archimedes discovered that a sphere's volume is exactly $2/3$ the volume of its circumscribed cylinder, a result he considered his greatest achievement.