Volume of Spherical Cone

volume bounded by a right-circular central-cone, of half-angle alpha, and a sphere of radius rho

Jacobian

Find the Jacobian. The transformation from the spherical to the Cartesian coordinates is:

• x = rho sin(alpha) cos(theta)
• y = rho sin(alpha) sin(theta)
• z = rho cos(alpha)

Then the Jacobian is the determinant of the partial derivatives:

J = det(

• dx / drho, dx / dalpha, dx / dtheta
• dy / drho, dy / dalpha, dy / dtheta
• dz / drho, dz / dalpha, dz / dtheta) )

= det(

• sin(alpha) cos(theta),   rho cos(alpha) cos(theta),    -rho sin(alpha) sin(theta)
• sin(alpha) sin(theta),    rho cos(alpha) sin(theta),       rho sin(alpha) cos(theta)
• cos(alpha),                -rho sin(alpha),                       0 )

= rho^2 ( sin(alpha) ((cos(alpha))^2 + (sin(alpha))^2) ((cos(theta))^2 + (sin(theta))^2) )

= rho^2 sin(alpha)

spherical cone

A spherical cone has its apex at the center of the sphere, of radius a, and its base is a region on the surface of the sphere.  The volume bounded by this spherical cone is given by the triple-integral

• integral, from 0 to alpha, of integral, from 0 to 2pi, of integral, from 0 to a, of rho^2 sin(alpha) drho dtheta dalpha.

For this cone, the integral separates into an area-integral times the integral with respect to rho

• integral, from 0 to alpha, of integral, from 0 to 2pi, of sin(alpha) dtheta dalpha * integral, from 0 to a, of rho^2 drho =
• = area of the spherical cap on a unit-sphere) * a^3 / 3 =
• = (area of the spherical cap) * a / 3.

circular cap

We let x be the thickness of this cap

• x = a (1 - cos(alpha)).

Then, we have

• cos(alpha) = (a - x) / a
• (sin(alpha))^2 = x (2 a - x) / a^2.

The area of a circular cap is given by the double-integral

• A of cap = integral, from 0 to 2pi, of integral, from 0 to alpha, of a^2 sin(alpha) dalpha dtheta.

This integral separates into the product of two single-integrals

• A of cap = integral, from 0 to 2 pi a^2, of dtheta * integral, from 0 to alpha, of sin(alpha) dalpha =
• = 2 pi a^2 * from 0 to alpha of (- cos(alpha)) =
• = 2 pi a^2 (1 - cos(alpha)) =
• = 2 pi a x.

The volume bounded by a circular cap is given by the integral

• V of cap = pi a^3 integral, from 0 to alpha, of (sin(alpha))^2 sin(alpha) dalpha =
• = pi a^3 integral, from 0 to alpha, of (1 - (cos(alpha))^2) sin(alpha) dalpha =
• = pi a^3 from 0 to alpha of (- cos(alpha) + (cos(alpha))^3 / 3) =
• = pi a^3 (2 - 3 cos(alpha) + (cos(alpha))^3) / 3 =
• = pi (2 a^3 - 3 a^2 (a - x) + (a - x)^3) =
• = pi x^2 (3 a - x) / 3.

The volume of a flat-based circular cone of course is

• V of flat cone = pi a^3 (sin(alpha))^2 cos(alpha) / 3 =
• = pi x (2 a - x) (a - x) / 3.

caped cone

Now, we have two ways of computing the volume bounded by a caped cone.  They have to yield the same results.

sum

We add the volume bounded by the cap plus that bounded by the flat cone

in terms of alpha

• pi a^3 (2 - 3 cos(alpha) + (cos(alpha))^3) / 3 + pi a^3 (sin(alpha))^2 cos(alpha) / 3 =
• = 2 pi a^3 (1 - cos(alpha)) / 3.

in terms of x

• pi x^2 (3 a - x) / 3 + pi x (2 a - x) (a - x) / 3 =
• = 2 pi a^2 x / 3.

product

in terms of alpha

• 2 pi a^2 (1 - cos(alpha)) * a / 3 =
• = 2 pi a^3 (1 - cos(alpha)) / 3

in terms of x

• 2 pi a x * a / 3 =
• = 2 pi a^2 x./ 3.

Copyright (c) 1999, 2000 by R. I. 'Scibor-Marchocki

last modified Sunday 03-rd December 2000.