volume of spherical cap and spherical sector
Proof. The sphere may be formed by letting the circle , i.e. , rotate about the -axis. Let the spherical cap be the portion from the sphere on the left of the plane at perpendicular to the -axis.
Then the for the volume of solid of revolution yields the volume in question:
Theorem 2. The volume of a spherical sector is , where is the height of the spherical cap of the spherical sector and is the radius of the sphere.
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 is
Substituting this to the expression of and simplifying give , Q.E.D.
|Title||volume of spherical cap and spherical sector|
|Date of creation||2013-03-22 18:19:14|
|Last modified on||2013-03-22 18:19:14|
|Last modified by||pahio (2872)|
|Synonym||volume of spherical cap|
|Synonym||volume of spherical sector|