Let A, B, and C be points on a sphere with center O and radius r. Angles AOB, BOC, and COA are 60 degrees. What is the volume of the interior of the sphere between the planes AOB, BOC, and COA?
1. "RE: Volume Question - My Bad"
In response to message #0
>Can somebody help me with this problem: > >Let A, B, and C be points on a sphere with center O and >radius r. >Angles AOB, BOC, and COA are 60 degrees. What is the volume >of the interior of the sphere between the planes AOB, BOC, >and COA?
I should have said: What is the volume bounded by the planes AOB, BOC, and COA and the interior of the spherical triangle ABC?
2. "RE: Volume Question - My Bad"
In response to message #1
What you have to do is this.
First find the angles between the planes forming the sides of a regular tetrahedron. They are all the same and are twice the angle whose sin is sqrt(3)/3. Your planes determine an equiangular spherical triangle with angles equal to the one you just found. On a unit'sphere, the area of a spherical triangle is the excess of the sum of angles over p. The volume of the "spherical pyramid" you are looking for is 1/3 of the base (spherical) triangle area times the radius.
3. "RE: Volume Question - My Bad"
In response to message #2
Thanks Alex. I dug up my "VNR Concise Encyclopedia of Mathematics" and found the formulas for the area of the spherical triangle. But, I was not able to find anything on the volume of the "spherical pyramid". Got any references?
4. "RE: Volume Question - My Bad"
In response to message #3
This is rather obvious. The area of the sphere of radius 1 is 4p. It's volume is 4p/3. The volume of a spherical pyramid is to the volume of the sphere as the area of its base to the area of the sphere.
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