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Gear300

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In summary, for a sphere, the relation between steradians and the area they cover is O = A/(r^2), where O is the measure of the solid angle, A is the area it covers, and r is the radius. If only given the half-angle of the steradian, the area can be found by integrating over the sphere using spherical coordinates. The region corresponds to the range 0\leq \phi < 2 \pi and 0 \leq \theta < \theta_0, where \theta_0 is the half angle. The area can be evaluated using the integral A = \int_0^{2 \pi} d \phi \int_0^{\theta_0} d

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Gear300

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- #2

StatusX

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[tex] \int_0^{2 \pi} d \phi \int_0^\pi d\theta R^2 \sin\theta [/tex]

The region you're talking about then corresponds to the range [itex]0\leq \phi < 2 \pi [/itex] and [itex]0 \leq \theta < \theta_0[/itex], where [itex]\theta_0[/itex] is the half angle. So you can find the area by evaluating:

[tex] A = \int_0^{2 \pi} d \phi \int_0^{\theta_0} d\theta R^2 \sin\theta [/tex]

- #3

Gear300

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...wait...wut? I sort of half get what you're saying. I was using earlier the integral with the sine in it...but where did the integral of dphi pop out of?

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- #4

StatusX

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Look up spherical coordinates.

The formula for finding the area of a portion of a sphere is A = 2πr²(h - r), where A is the area, r is the radius of the sphere, and h is the height of the portion.

The height of a portion of a sphere can be calculated by taking the difference between the radius of the sphere and the distance from the center of the sphere to the top of the portion.

Yes, the area of a portion of a sphere can be calculated using the formula A = 2πr²(1 - cosθ), where θ is the angle of the portion in radians.

As the height of a portion of a sphere increases, the area also increases. This is because the height directly affects the surface area of the portion, which is a factor in the formula for finding the area.

The surface area of a portion of a sphere is a fraction of the surface area of the entire sphere. This fraction is calculated by taking the ratio of the height of the portion to the radius of the sphere.

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