volume of spherical cap and spherical sector

Theorem 1.  The volume of a spherical cap is  $\pi h^{2}\!\left(r\!-\!\frac{h}{3}\right)$,  when $h$ is its height and $r$ is the radius of the sphere.

Proof.  The sphere may be formed by letting the circle  $(x\!-\!r)^{2}\!+\!y^{2}=r^{2}$,  i.e.  $y=(\pm)\sqrt{rx\!-\!x^{2}}$,  rotate about the $x$-axis.  Let the spherical cap be the portion from the sphere on the left of the plane at  $x=h$perpendicular to the $x$-axis.

Then the for the volume of solid of revolution yields the volume in question:

 $V=\pi\!\int_{0}^{h}(\sqrt{rx\!-\!x^{2}})^{2}\,dx=\pi\!\int_{0}^{h}(2rx\!-\!x^{% 2})\,dx=\pi\!\operatornamewithlimits{\Big{/}}_{\!\!\!x=0}^{\,\quad h}\left(rx^% {2}\!-\!\frac{x^{3}}{3}\right)=\pi{h}^{2}\!\left(r\!-\!\frac{h}{3}\right).\\$

Theorem 2.  The volume of a spherical sector is  $\frac{2}{3}\pi{r}^{2}h$,  where $h$ is the height of the spherical cap of the spherical sector and $r$ is the radius of the sphere.

Proof.  The volume $V$ of the spherical sector equals to the sum or difference of the spherical cap and the circular cone depending on whether  $h  or  $h>r$.  If the radius of the base circle of the cone is $\varrho$, then

 $V=\begin{cases}\pi{h}^{2}(r\!-\!\frac{h}{3})+\frac{1}{3}\pi{\varrho}^{2}(r\!-% \!h)&\mbox{when\, hr.}\end{cases}$

But one can see that both expressions of $V$ are identical.  Moreover, if $c$ is the great circle of the sphere having as a diameter the line of the axis of the cone and if $P$ is the midpoint of the base of the cone, then in both cases, the power of the point $P$ with respect to the circle $c$ is

 $\varrho^{2}=(2r\!-\!h)h.$

Substituting this to the expression of $V$ and simplifying give  $V=\frac{2}{3}\pi{r}^{2}h$,  Q.E.D.

 Title volume of spherical cap and spherical sector Canonical name VolumeOfSphericalCapAndSphericalSector Date of creation 2013-03-22 18:19:14 Last modified on 2013-03-22 18:19:14 Owner pahio (2872) Last modified by pahio (2872) Numerical id 7 Author pahio (2872) Entry type Theorem Classification msc 26B15 Classification msc 53A05 Classification msc 51M04 Synonym volume of spherical cap Synonym volume of spherical sector Related topic SubstitutionNotation Related topic GreatCircle Related topic Diameter2 Related topic PowerOfPoint