Volume

Volume measures three-dimensional capacity in cubic units and connects base area, height, scale, and real containers.

A multiple integral accumulates small area or volume elements over a region. A translucent solid is divided into representative elliptical slices. The repeated cross-sections show how a volume can be accumulated one thin slice at a time.

Volume measures the amount of three-dimensional space inside a solid and is expressed in cubic units.

Why units are cubed

A cubic centimetre is a $1$ cm by $1$ cm by $1$ cm cube. A rectangular prism with dimensions $4$, $3$, and $2$ cm can be filled by $4\times3\times2=24$ such cubes, so its volume is $24$ cm³.

Volume describes capacity or occupied space; surface area describes the material covering the outside.

Prisms and cylinders

Any prism has constant cross-section, so

$$ V=Bh, $$

where $B$ is base area and $h$ is perpendicular length. For a rectangular prism, this becomes $V=lwh$.

A cylinder has circular base area $\pi r^2$:

V=\pi r^2h.

Pyramids and cones

A pyramid or cone with the same base and perpendicular height as a prism or cylinder has one third the volume:

V=\frac13Bh.

For a cone, $V=\tfrac13\pi r^2h$. Use vertical height, not slant height, because the formula measures perpendicular stacking from base to tip.

Spheres

A sphere of radius $r$ has volume

V=\frac43\pi r^3.

Because radius is cubed, a modest increase in radius produces a much larger change in volume. Doubling radius multiplies volume by $2^3=8$.

Common mistakes

Using diameter as radius. Divide diameter by $2$ before squaring or cubing.

Forgetting the factor $1/3$. Pyramids and cones are one third of matching prisms or cylinders.

Using slant height. Volume uses perpendicular height.

Mixing units. Convert all dimensions to a common unit first.

Reporting square units. Volume requires cubic units.

Quick self-check

What is the base, and what is its area?
Is height perpendicular to the base?
Is the solid prism-like, pyramid-like, spherical, or composite?
Do the units and scale of the answer make sense?