Mathematical expressions have been derived for finding out the solid angle subtended by the rotatory solids like Ellipsoid, Sphere & Paraboloid by using the concept of an imaginary right conical surface completely enclosing the rotatory solids. This concept makes the derivations very easy.

1. Nov, 2013
M.M.M. University of Technology, Gorakhpur-273010 (UP) India
1. The solid angle subtended by an ellipsoid generated by rotating an ellipse ⁄ ⁄ about the major axis at any point lying on the major axis at a distance d from the centre is given as
( √ )
Consider an ellipsoid generated by rotating an ellipse with major & minor axes 2a & 2b respectively about the major axis & a given point say P lying on the major axis (x-axis) at a distance d from the centre O (origin). (see the fig1. below)
Fig1: Solid angle subtended by an ellipsoid at a point lying on the major axis
Now consider an imaginary right conical surface with minimum apex angle & apex at the given point P such that it completely encloses the ellipsoid. We know the equation of generating curve (i.e. ellipse)
Now draw two tangents from the point as PS & PR on the ellipse & let R be any parametric point on the ellipse. Equation of line PR is given as

2. ( )
But line PR is tangent to the ellipse hence applying the condition of tangency to the ellipse for general line as follows ⇒ ( ) ( ) ⇒ ⇒ ⇒ √ √
Now, co-ordinates of point R ( √ ) ⇒ √
In right ⇒ √ ( ) √( ) ( √ ) √
Since, the ellipsoid is completely enclosed by the imaginary conical surface with apex point P ⇒ ( √ )

3. 2. Similarly, it can be proved that the solid angle subtended by an ellipsoid generated by rotating an ellipse ⁄ ⁄ about the minor axis at any point lying on the minor axis at a distance d from the centre is given as
( √ )
3. Solid angle subtended by a sphere with radius R at any point lying at a distance d from the centre
It can be obtained by setting in any of the above results of an ellipsoid we get ( √ )
4. Solid angle subtended by a paraboloid generated by rotating the parabola about the axis at any point lying on the axis at a distance d from the vertex is given as
( √ )
Let there be a paraboloid which is generated by rotating the parabola about its axis (x-axis) & any given point P at a distance d from the vertex O (origin)
Now consider an imaginary right conical surface with minimum apex angle & apex at the given point P such that it completely encloses the paraboloid. We know the equation of generating curve (i.e. parabola)
Now draw two tangents from the point as PS & PR on the parabola & let R be any parametric point on the parabola. (see the fig 2.) ⇒
In right ⇒
But the slope of the tangent PR to the parabola at the point is obtained by differentiating the equation w.r.t.

4. Fig 2: Solid angle subtended by a paraboloid at a point lying on the axis ( ) ( )
On setting the value of in the eq(I), we have ⇒ √ √ √ √ (√ ) √
Since, the paraboloid is completely enclosed by the imaginary conical surface with apex point P ⇒ ( √ )
Note: Above derivations are done by the concept of right cone. ‘Solid angle subtended by the right circular cone at the apex point’ has directly been taken from author’s book “Advanced Geometry by H.C. Rajpoot” which has its derivation & detailed explanation.