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volume of spherical cap and spherical sector
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(Theorem)
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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: $$V = \pi\!\int_0^h(\sqrt{rx\!-\!x^2})^2\,dx = \pi\!\int_0^h(2rx\!-\!x^2)\,dx = \pi\!\sijoitus{x=0}{\quad h}\left(rx^2\!-\!\frac{x^3}{3}\right) = \pi{h}^2\!\left(r\!-\!\frac{h}{3}\right).\\$$
Theorem 2. The volume of a spherical sector is $\frac{2}{3}\pi{r}^2h$ , 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
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 $V = \frac{2}{3}\pi{r}^2h$ , Q.E.D.
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"volume of spherical cap and spherical sector" is owned by pahio.
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Cross-references: volume of solid of revolution, perpendicular, plane, rotate, circle, proof, sphere, radius, spherical cap, volume, theorem
There is 1 reference to this entry.
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 2955 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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Pending Errata and Addenda
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