volume of spherical cap and spherical sector

Theorem 1.  The volume of a spherical capMathworldPlanetmath is  πh2(r-h3),  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+y2=r2,  i.e.  y=(±)rx-x2,  rotate about the x-axis.  Let the spherical cap be the portion from the sphere on the left of the plane at  x=hperpendicularPlanetmathPlanetmathPlanetmath to the x-axis.

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


Theorem 2.  The volume of a spherical sector is  23πr2h,  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 ϱ, then

V={πh2(r-h3)+13πϱ2(r-h)when h<r,πh2(r-h3)-13πϱ2(h-r)when h>r.

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


Substituting this to the expression of V and simplifying give  V=23πr2h,  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