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/
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