HCR's Method of concentric cones is the simplest & most versatile method to find out the solid angle subtended by a torus at any point lying on its geometrical axis.

1. Copyright© Harish Chandra Rajpoot 2014
Mr Harish Chandra Rajpoot
M.M.M. University of Technology, Gorakhpur-273010 (UP), India 16/11/2014
“Solid angle subtended by a torus at any point lying on its geometrical axis i.e. normal axis passing
through its centre”
Let there be a torus with inner & outer radii ‘r’ & ‘R’ respectively & a given point say ( ) lying on the axis
at a normal height ‘h’ from the centre ‘O’ (point ( ) is lying perpendicularly outwards to the plane of
paper at a height h from the centre O, as shown by top view in the figure 1 below)
Figure 1: Two imaginary concentric right conical surfaces, inner PB & outer PA, with apex
angles respectively encloses the torus completely (as shown by the dotted
lines in the front view (sectional view))
Front View
(Sectional view of torus)
Top View
(Point ( ) is lying
normally outwards to the
plane of paper at a height h
from the centre O)
Now, draw the tangents PA & PB from the point ( ) at the surface of torus (upper part of figure 1 is
showing the sectional view (front view) of torus)
⇒

2. Copyright© Harish Chandra Rajpoot 2014
⇒
In right
⇒ √ √( ) ( ) √( )
⇒ ⇒
( )
( √( ) ) √( )
In right
⇒ ⇒
( )
( √( ) ) √( )
Concentric cones: These are the cones (or conical surfaces) having same vertex (apex point) & the geometrical
axis and the equal vertical height. Dotted lines PB & PA are showing inner & outer conical surfaces respectively
in top view in above figure 1.
We know that the solid angle subtended by any cone with apex angle at its apex point is given as
( ).
Now, let’s consider two imaginary concentric hollow right cones (i.e. inner conical surface PB & outer conical
surface PA) having common apex at the given point ( ), equal normal height , minimum apex
angle & maximum apex angle respectively such that the torus is completely
covered by both the conical surfaces i.e. outer PA & inner PB.
In such case, since the torus is completely covered by inner & outer conical surfaces hence the solid angle
subtended by the torus will be equal to the solid angle subtended by the conical annular-space or equal to the
difference of solid angles subtended by the outer & inner conical surfaces respectively. Hence, the solid angle
( ) subtended by the torus at the given point ‘P’ lying on the geometrical axis
( ( ) ( ( )
⇒ ( ) ( )
( ( )) ( ( ))
( ( ) ( ))
( )
On setting the values of in the above expression, we get
⇒
√( ) √( )

3. Copyright© Harish Chandra Rajpoot 2014
( )( )
( ) ( )
⇒ (
( )
) [ ) ( )
Solid angle subtended by a torus at its centre
As shown in the figure 2 below
In this case, solid angle subtended by the torus is obtained by setting
as follows
(
( ) ( )
)
(
( )
)
( )( )
( )
( )
⇒ ( ) ( )
Important deductions:
1. On setting in the above eq(1) or (2), we get
(
( )
)
( )
It is clear from the above result that solid angle subtended a torus with diameter i.e. circle at any
point on its axis is always zero.
2. It is obvious from the eq(1) that the solid angle subtended by a torus with finite radii at any point lying at the
infinity tends to zero i.e. ⇒
(
( )
)
Solid Angle subtended by a Simply Closed Torus at any point lying on its vertical axis
A torus with zero inner radius ( ) is called a simply closed torus or
It can be defined as a solid or a surface generated simply by rotating a circle about its tangent at any point.
(As shown in the figure 3 below) In order to calculate the solid angle subtended by a torus at any point
( ) lying on the vertical axis at a height from the centre O of the torus, let’s apply the conditions in the
eq (1) as in the following cases
Figure 2: Solid angle subtended by the torus at
its centre O

4. Copyright© Harish Chandra Rajpoot 2014
1. Solid angle subtended by a simply closed torus (with zero inner radius
( )) at any point lying at a normal height on the geometrical/normal
axis is obtained by setting in the eq(1) as follows
(
( )
( )
) ( )
( ) ( )
2. Solid angle subtended by a simply closed torus at its centre is obtained by
setting in the eq(1) as follows
(
( )
( ) ( )
) ( )
The above result is absolutely correct which shows that the centre of any
torus having zero inner radius ( ) is completely surrounded by it hence a
simply closed torus subtends a solid angle of at its centre.
Note: Above articles had been derived & illustrated by Mr H.C. Rajpoot (B Tech, Mechanical Engineering)
M.M.M. University of Technology, Gorakhpur-273010 (UP) India Nov, 2014
Email: rajpootharishchandra@gmail.com
Author’s Home Page: https://notionpress.com/author/HarishChandraRajpoot
Courtesy: Advanced Geometry by Harish Chandra Rajpoot
Copyright© Harish Chandra Rajpoot 2014
Figure 3: A simply closed torus having
inner radius zero