PlanetMath: volume of spherical cap and spherical sector

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volume of spherical cap and spherical sector

(Theorem)

Theorem 1. The volume of a spherical cap is $\pi h^2\!\left(r\!-\!\frac{h}{3}\right)$ , when is its height and 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 -axis. Let the spherical cap be the portion cut from the sphere on the left of the plane at perpendicular to the -axis.

$\begin{pspicture}(-0.8,-3.5)(7,3.5) \psdots(0,0)(1.2,0)(3,0)(5.95,0) \rput(-0.2,... ...0,0) \psline{->}(6,0)(6.7,0) \rput(-0.8,-3.5){.} \rput(7,3.5){.} \end{pspicture}$

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

$\displaystyle V = \pi\!\int_0^h(\sqrt{rx\!-\!x^2})^2\,dx = \pi\!\int_0^h(2rx\!-... ...ft(rx^2\!-\!\frac{x^3}{3}\right) = \pi{h}^2\!\left(r\!-\!\frac{h}{3}\right).\\$

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

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

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

But one can see that both expressions of

are identical. Moreover, if

is the great circle of the sphere having as a diameter the line of the axis of the cone and if

is the midpoint of the base of the cone, then in both cases, the power of the point

with respect to the circle

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

Substituting this to the expression of

and simplifying give $V = \frac{2}{3}\pi{r}^2h$ , Q.E.D.

"volume of spherical cap and spherical sector" is owned by pahio.

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volume of spherical cap, volume of spherical sector

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Cross-references: volume of solid of revolution, perpendicular, plane, rotate, circle, proof, sphere, radius, spherical cap, volume

This is version 4 of volume of spherical cap and spherical sector, born on 2008-08-16, modified 2008-08-18.
Object id is 10950, canonical name is VolumeOfSphericalCapAndSphericalSector.
Accessed 871 times total.

Classification:

AMS MSC:	51M04 (Geometry :: Real and complex geometry :: Elementary problems in Euclidean geometries)
	53A05 (Differential geometry :: Classical differential geometry :: Surfaces in Euclidean space)
	26B15 (Real functions :: Functions of several variables :: Integration: length, area, volume)

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