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The Relationship Between the
Solid Angle of a Conical Point
and Its Angular Deficit

The solid angle and the angular deficit do not measure the same thing, but there is a relationship between them. It is an antithetical relation. When the solid angle
is 2π the angular deficit is zero. When the angular deficit is 2π the solid anglle is zero.

Let γ be half the apex angle of a cone. On a sphere of unit radius the cone will subtend an area equal to
2π(1 − cos(γ)). This is the solid angle of the cone, which will be denoted as Θ.

The radius of the circle of intersection of the cone and the unit sphere is equal
to sin(γ). The length of the circumference of that circle is then 2πsin(γ) but the circumference of
the cicle within which the unrolled cone lies is 2π. Thus the angular deficit α of the cone isgiven by