Degrees in a Sphere? Steradians

Date: 09/27/2001 at 23:49:44
From: Caleigh Wright
Subject: Circles vs.spheres

If one can say that a circle contains 360 degrees, how many degrees, 
then, can one say are in a sphere? Or is this proposterous?

Date: 09/28/2001 at 15:40:53
From: Doctor Rick
Subject: Re: Circles vs.spheres

Hi, Caleigh.

We can't say how many degrees there are in a sphere, any more than we 
can say how many feet there are in an acre. Feet are a measure of 
length, and an acre is an area, not a length. You can't measure an 
area with a tape measure. Likewise, degrees are a measure of an angle; 
you can sweep out a circle by swinging a line through an angle of 360 
degrees. But you can't sweep out a sphere by swinging a line through 
some angle, so angle measure won't do to measure a sphere.

Let's think more about the analogy to length versus area. We can 
measure area in *square* feet. Is there anything like "square degrees" 
that we can use to measure a sphere? Yes, there is! But instead of 
degrees, we start with radians, a different measure of angles. We come 
up with something that could perhaps be called "square radians." 
Squares won't really enter into it, though, so instead we call the 
unit a "steradian" (like "stereo radian"; stereo is from the Greek for 
solid, or 3-dimensional). We say that it is a measure of "solid 
angle."

Do you know the idea behind the radian measure of an angle? You draw a 
circle using the vertex of the angle as center. Then measure the 
length of the arc cut off by the two legs of the angle, and divide 
this length by the radius of the circle. The ratio is the same no 
matter what size circle you draw; we call the ratio the radian measure 
of the angle.

If you do this with a full circle (a 360-degree angle), then the arc 
is the full circumference of the circle. Its length is 2 pi times the 
radius of the circle. Divide this by the radius, and you get 2 pi. 
Thus 360 degrees equal 2 pi (approximately 6.28) radians.

What is a solid angle? One way to picture a solid angle is the tip of 
a cone or a pyramid. A tall narrow cone has a small solid angle at the 
tip; a broad flat cone has a large solid angle at the tip. The solid 
angle doesn't have to be "round" though. Just as you can have 
different shapes with the same area, you can have solid angles with 
different "shapes" but the same measure (in steradians). For instance, 
the peak of a triangular pyramid is sort of a "triangular" solid 
angle, and the peak of a square pyramid is sort of a "square" solid 
angle.

I have shown you how the measure of an angle is related to the length 
of an arc. Now let's think about a sphere and a "solid angle." Take 
the peak of that cone or pyramid, and draw a sphere around it. (You'll 
have to imagine this; I can't draw in the air.) The solid angle cuts 
off a piece of the sphere. If we measure the area of this piece, and 
divide the area by the square of the radius of the sphere, then we 
have a measure of the solid angle in steradians.

The surface area of a sphere is 4 pi times the square of the radius. 
Therefore the entire sphere has a solid angle of 4 pi steradians. 
That's as close as we're going to get to an answer for your question: 
how many degrees are there in a sphere?

We have a few discussions of solid angles and steradians in our Dr. 
Math archives, written for those with a higher level of math 
(trigonometry and calculus).

   Measuring Angles Using Steradians
   http://mathforum.org/dr.math/problems/scaffidi2.8.96.html   

   Use of Steradians
   http://mathforum.org/dr.math/problems/skaffml.7.26.96.html   

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/