# How to Find the Area Covered by a Solid Angle Using the Half-Angle?

• Gear300
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
Gear300
For a sphere, the relation between steradians and the area they cover is O = A/(r^2), in which O is the measure of the solid angle, A is the area it covers, and r is the radius. If I were instead given the half-angle of the steradian...meaning that if there was a central axis running through the solid angle, connecting the surface of the sphere to its center, the half-angle would simply be the angle between the central axis and the edge of the solid angle...then how would I find the area covered by the solid angle in respect to the half angle.

You can integrate over the sphere using spherical coordinates by:

$$\int_0^{2 \pi} d \phi \int_0^\pi d\theta R^2 \sin\theta$$

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

$$A = \int_0^{2 \pi} d \phi \int_0^{\theta_0} d\theta R^2 \sin\theta$$

...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?

Last edited:
Look up spherical coordinates.

## 1. What is the formula for finding the area of a portion of a sphere?

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.

## 2. How do you calculate the height of a portion of a sphere?

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.

## 3. Can you find the area of a portion of a sphere without knowing the height?

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.

## 4. How does the area of a portion of a sphere change as the height increases?

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.

## 5. How is the area of a portion of a sphere related to the surface area of the entire sphere?

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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