Lesson Introduction to Solid angles

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In this Lesson we will understand the concept of solid angles and their units. We will also look into some common figures and how to calculate solid angles for them.

Introduction to Solid Angles

In simple terms the solid angle is that fraction of the surface of a <A HREF=Sphere.wikipedia>sphere</A> that a particular object covers, as seen by an observer at the sphere's center.

{{{drawing(160,160,-15,15,-15,15,circle(0,0,15),circle(2,6,2),line(0,0,3.6,4.8),line(0,0,0.01,5.73),locate(5,8.5,O[1]),locate(0,0,O))}}}

As shown here in figure, for a observer standing at sphere center O will see a curve ( here it is a <A HREF=Circle.wikipedia>circle</A> of center {{{O[1]}}}) being formed on the surface of the sphere. The ratio of this <A HREF=http://www.algebra.com/algebra/homework/Surface-area/Lessons.html>surface area</A> observed to square of radius of sphere will give the solid angle.

By definition the solid angle is described as an angle that when seen from the center of a sphere, includes a given area on the surface of that sphere. The value of the solid angle is numerically equal to the size of that area divided by the square of the radius of the sphere. It is mathematically denoted by "Omega".

{{{Solid Angle= Omega = k S/R^2}}}

where,

k is the proportionality constant,usually taken as 1.
S is the surface area of projection onto the sphere.
and, R is the radius of the sphere.

Units of Solid Angle

Mathematically, the solid angle is unitless, but for practical reasons, the steradian is assigned. Standard unit of a solid angle is the Steradian (sr).The solid angle is often a function of direction.

1 steradian can be defined as, for a sphere with a radius of 1 meter. A cone that covers an area of 1{{{m^2}}} on the surface of the sphere encloses a solid angle of 1 steradian. A full sphere has a solid angle of {{{4*pi}}} steradian.

Solid angles for common objects

<A HREF=http://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Cone.wikipedia>Cone</A>, spherical cap, hemisphere

{{{drawing(160,160,-10,10,-10,10,locate(0,2,O),circle(0,0,10),line(0,0,-7.07,-7.07),line(0,0,7.07,-7.07),line(-7.07,-7.07,7.07,-7.07),locate(0,-1,a))}}}

For an observer at center of the sphere a cone will be formed along his line of view. The solid angle will be calculated by subtracting area of cone from <A HREF=Circle-sector.wikipedia>area of sector</A> forming angle a at the center.The difference in area is called the spherical cap of a sphere.

For a sphere of radius 1 the solid angle is, {{{2*pi*(1-cos(a/2))}}}.

When {{{a = pi}}}, the spherical cap becomes a hemisphere having a solid angle {{{2pi}}}.

Pyramid

The solid angle of a four-sided right rectangular pyramid with apex angles a and b (measured to the faces of the pyramid) is {{{4arcsin(sin(a/2)sin(b/2))}}}

To read more on solid angle refer to articles on <A HREF=http://www.algebra.com/algebra/homework/Angles/solid_angle.wikipedia>wikipedia</A>