My Math Forum Area of sphere inside cube

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 March 27th, 2013, 10:43 AM #1 Newbie   Joined: Mar 2013 Posts: 2 Thanks: 0 Area of sphere inside cube Ok, so a cube and a sphere are both centered at origo. The cube has side lengths L. I need to know the surface area of the part of the sphere that is inside the cube, for all possible r. There will be three different equations, one for $0 < r < L/2$, when the entire sphere is inside the cube, one for $L/2 < r < L/\sqrt(2)$, when six spherical caps of the sphere stick out of the cube sides , and one for $L/\sqrt(2) < r < L\sqrt(3) / 2$, when only the corners of the cube is visible. I have figured out the two first, but not the last. Any help would be greatly appreciated!
 March 27th, 2013, 12:41 PM #2 Senior Member   Joined: Feb 2012 Posts: 628 Thanks: 1 Re: Area of sphere inside cube Calculate the cross-sectional area for the smaller part of the sphere cut by the plane which coincides with one of the sides of the cube, and call this area C. Now, imagine six of these cross-sectional areas, one for each side of the cube. Let the area divided by two adjacent cross-sections along an edge be E, the area within any one cross-section not shared by the others F, and the area of the sphere not covered by these cross-sections be X. Now, we want to calculate X, since the area of the sphere not covered by these cross-sections is the area of the sphere which is inside the cube. Letting A represent the surface area of the whole sphere, we have the following equations: 6C = 24E + 6F (since there are 6 faces and 12 edges but each edge section is counted twice) A = 6F + 12E + X 6C - A = 12E - X Now, if you can calculate E, then you can calculate X. I don't know how hard it would be to calculate E though.
March 27th, 2013, 02:09 PM   #3
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Re: Area of sphere inside cube

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 Originally Posted by icemanfan Now, if you can calculate E, then you can calculate X. I don't know how hard it would be to calculate E though.
This is a good start, thanks E is kind of like a spherical lune, except it is an intersection between the sphere and two SMALL circles, not two large. The two small circles are perpendicular to each other. It could be calculated as 2 times the area of the sphere bounded by one great circle and one small, the small having a distance L/2 from the center and a 45 degree angle to the great circle. Anyone know how to calculate such an area?

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