## You are here

Homevolume of spherical cap and spherical sector

## Primary tabs

# 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 cut from the sphere on the left of the plane at $x=h$ perpendicular to the $x$-axis.

Then the formula 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 $h<r$ 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\, $h<r$,}\\ \pi{h}^{2}(r\!-\!\frac{h}{3})-\frac{1}{3}\pi{\varrho}^{2}(h\!-\!r)&\mbox{when% \, $h>r$.}\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.

## Mathematics Subject Classification

26B15*no label found*53A05

*no label found*51M04

*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections

## Comments

## Anomaly in the autolinking

Hi, what may be the cause that the autolinking doesn't work in theorem 2 and its proof (http://planetmath.org/?op=getobj&from=objects&id=10950)? E.g. the words "spherical sector", "great circle", "diameter", "power of point" don't link normally.

Jussi