Volume Formulas

Side cubed. A cube of side $s$ tiles space with $s^3$ unit cubes — the 3D analogue of the unit-square argument.

Length × width × height. Base area is $l w$; stacking $h$ layers of that base gives $lwh$.

Base area times height. By Cavalieri’s principle, any prism with the same cross-section and height has the same volume — so triangular, hexagonal, oblique prisms all use this single formula.

One-third of the corresponding prism. The "one-third" comes from integrating $A_{\text{base}}\bigl(\tfrac{z}{h}\bigr)^2$ from 0 to $h$ — the cross-section shrinks linearly.

Same one-third rule as the pyramid, with circular base $\pi r^2$. Three cones with the same base and height fill exactly one cylinder.

Two parallel circular faces of radii $R$ (bottom) and $r$ (top), height $h$. Derive by subtracting the small cone from the big cone; the cross-term $Rr$ comes from the difference of cubes.

Special case of the general prism: circular base $\pi r^2$ stacked to height $h$. Oblique cylinders use the same formula thanks to Cavalieri.

Outer cylinder volume minus inner cylinder volume — the same subtraction trick as the annulus extended in the third dimension.

The famous "four-thirds pi r-cubed." Archimedes’ result: a sphere is exactly $\tfrac{2}{3}$ of the smallest cylinder that contains it.

Half of a sphere — exactly half of $\tfrac{4}{3}\pi r^3$. Useful for domes, bowls, and integration setups.

Three semi-axes $a, b, c$. When $a = b = c = r$ you recover the sphere $\tfrac{4}{3}\pi r^3$ — a sphere is a special ellipsoid.

Major radius $R$ (center to tube center), minor radius $r$ (tube). Pappus theorem: area $\pi r^2$ swept around a circle of circumference $2\pi R$.