im fairly sure theres a way to solve this by useing a complex plane and some progective geomatry.
How Many Degrees In A Sphere?
Posts (131)

#742828  14 years ago

#742839  14 years ago
In reply to randainium3, #26:
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 3dimensional). 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 360degree 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? 
#742859  14 years ago
In reply to TheDarksage, #27:
Feet can be used as the circumfrence of a circle, much like degrees, only degrees do not need the radius.
For a circle, it is 2 * Pi * R.
For a sphere, it is 4 * Pi * R ^ 2.
If 2 * Pi * R = 360 degrees, then 4 * Pi * R ^ 2 = 360 degrees * (2 * R).
360 degrees * (2 * Pi * R) / Pi
360 degrees * 360 degrees / Pi
Results in: 129600 / Pi Degrees ^ 2
So a sphere has 129600 / Pi square degrees, or more simply 4 Pi steradians. Just like that website said. 
#742876  14 years ago
In reply to reedmon29, #28:
You can not use this formula for other round 3d objects though leading me to believe that it isn't true.
You can't simply say... 360x360=total degrees. For example the formula would not work for a cone with a 20x30 degree sweep. 
#742914  14 years ago
In reply to otto, #30:
How is a cone not a round 3d object? It is a series of circles creating a 3d object just like a sphere. 
#742924  14 years ago
In reply to otto, #34:
Technically, every shape and object comes to a point... :/ 
#742926  14 years ago
on it there would be basicly two circles on that is horzontal nd one that is vertical the two circles would need the same diameter with one circle u would rotate it so the edge of the circle thats rotating is touching the circle staying still if u move the circle by one degree each time then u would get 360 degrees

#742943  14 years ago
In reply to TheDarksage, #35:
Then stop calling them circles. You cant take thousands of years of mathmatical principles of circles and then say "Well technically none of this stuff is really true because there is no such thing as a round object" Everyone needs to stop making up math that suites their needs at the moment and pick one damnd set of rules to follow. That and use real math that explains the real natural universe. Not all this hypothetical math. 
#742949  14 years ago
In reply to otto, #37:
How about this... the "point" of a cone is actually a circle also with an infinitely small radius.... A cone and a sphere in my opinion are two of the same kind of 3d object... 
#742964  14 years ago
In reply to Vendor, #41:
Ok, here we go.
By stating that its your opinion you have completely discredited everything you have said. Math is not something of opinion, it is something of measurment. Math does not have room for opinion or guessing. In my opinion, circles where created by the white man to trick the indians into teaching them how to grow corn. My opinion has no research to back it up. I just pulled it out of my ass when I was writing this because it makes sense to me. If it makes sense to me it must be mathmatical fact and I should present it as such.
This is a math thread, not an opinion thread. If you have an opinion, show the research you have done to back it up. 
#742985  14 years ago
In reply to otto, #46:
I thought it was pretty obvious that it was collected from another site... in any case... math is not all about facts... it is about theory and conjectures... but whatever you want to say.
Prove to me that a point of a cone is not a circle with an infinitely small radius.