Let's say you have a cone inside a sphere, with the tip at the center of the sphere and the circular edge at the surface of the sphere. Given the angle of the cone slant (the "angle" in 2D between the two cone walls) how do you find the subtended area of the sphere?
The formula for surface area of a section of a sphere?
- Thread starter Howard
- Start date
I'd say to do a surface integration with the limits determined by the cone's intersection with the sphere.
Well, if the cone was a plane (angle = 180 degrees), then its dome would be a hemisphere, and the surface area of that would be half the surface area of the sphere.
I don't think its a linear relationship (angle of cone vs. percent of sphere's surface area). My guess is that it would be some sort of sinusoidal relationship.
If you had y = (- cos(x/2) + 1)/2, where x is the angle of the cone, then y would return the percent surface area of the dome compared to the sphere as a whole. If you plug in 180 degrees, it returns .5.
That's just a guess though.
I don't think its a linear relationship (angle of cone vs. percent of sphere's surface area). My guess is that it would be some sort of sinusoidal relationship.
If you had y = (- cos(x/2) + 1)/2, where x is the angle of the cone, then y would return the percent surface area of the dome compared to the sphere as a whole. If you plug in 180 degrees, it returns .5.
That's just a guess though.
I think I've figured it out, you inferior Canadian weiner
http://www.monolithic.com/construction/formulas.pdf
That has a pesky term h, which is solved for easily with some trignometry.
Doing a little algebra, I got S.A. = 2 Pi r^2 (1-cos (x/2))
That's the same as what I guessed earlier ((- cos(x/2) + 1)/2), after you multiply it out by the S.A. of a sphere (4 pi r^2).
http://www.monolithic.com/construction/formulas.pdf
That has a pesky term h, which is solved for easily with some trignometry.
Doing a little algebra, I got S.A. = 2 Pi r^2 (1-cos (x/2))
That's the same as what I guessed earlier ((- cos(x/2) + 1)/2), after you multiply it out by the S.A. of a sphere (4 pi r^2).
MikeyIs4Dcats
Lifer
- Nov 5, 2001
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