Cavalieri at work - prism and sphere

Let us see Cavalieri’s principle applied to two classic cases: the oblique prism and the sphere.

Example — Volume of the oblique prism

A right prism and an oblique prism with the same base and the same height have the same volume: at every height $z$ the two cross-sections are congruent to the base (and hence of equal area). By Cavalieri, $V_{\text{obliquo}} = V_{\text{retto}} = B\cdot h.$

Example — Volume of the sphere via Cavalieri (Archimedes' proof)

Let us compare the hemisphere of radius $R$ with the solid obtained from the right cylinder of radius $R$ and height $R$ by removing a right cone of radius $R$ and height $R$ (a cone “inverted” with respect to the cylinder). At height $z\in[0,R]$ from the base:

• Cross-section of the hemisphere: disc of radius $r_S = \sqrt{R^2-z^2}$, area $\pi(R^2-z^2)$.
• Cross-section of the cylinder minus cone: circular annulus with outer radius $R$ and inner radius $r_C = z$ (from the cone), area $\pi R^2 - \pi z^2 = \pi(R^2-z^2)$.

The areas coincide at every height $z$. By Cavalieri: $V_{\text{semisfera}} = V_{\text{cilindro}} - V_{\text{cono}} = \pi R^2\cdot R - \frac{1}{3}\pi R^2\cdot R = \frac{2}{3}\pi R^3,$ whence $V_{\text{sfera}} = \dfrac{4}{3}\pi R^3$, the same result that we shall obtain with the integral.

Links

Topics: Integrale
Concepts: Principio di cavalieri · Volume per sezioni
Methods: Cavalieri prisma
Skills: Dimostrare · Geometria sintetica
People: Archimede · Bonaventura Cavalieri