TheoremProved

Cavalieri's principle

Statement

If two solids in three-dimensional space lie between two parallel planes $z = a$ and $z = b$ , and every plane $z = t$ ( $a \le t \le b$ ) parallel to those planes intersects the two solids in cross-sections of equal area $A_1(t) = A_2(t)$ , then the two solids have equal volume: $V_1 = V_2$ .

Why is it true?

Imagine two identical stacks of thin coins of height $b - a$ . Even if you push one stack sideways into a tilted or curved shape, every horizontal slice at height $t$ still cuts through the exact same coin area $A_1(t) = A_2(t)$ , and sliding the coins horizontally never creates or destroys volume — so the total volume $V_1 = V_2$ is unchanged.

Proof sketch

By Fubini's theorem (or the cross-sectional volume formula in integral calculus), the volume of a solid bounded between $z = a$ and $z = b$ with integrable cross-sectional area $A_i(t)$ at height $z = t$ is the definite integral $V_i = \int_a^b A_i(t)\,dt$ . Since $A_1(t) = A_2(t)$ for every $t \in [a, b]$ , their integrals over $[a, b]$ are equal: $V_1 = \int_a^b A_1(t)\,dt = \int_a^b A_2(t)\,dt = V_2$ .

Topics that use this theorem

Polyhedra and their volumes Spheres, cylinders, cones Definite Integral

Related theorems

Volume and surface area of a sphere (Archimedes)Fundamental theorem of calculus

Step-by-step proofs

No step-by-step proof yet for this theorem.

References

Bonaventura Cavalieri (1635). Geometria indivisibilibus continuorum nova quadam ratione promota
Carl B. Boyer, Uta C. Merzbach (2011). A History of Mathematics