Question

How can you determine the diameter of the sun?

Answer 1

If `\theta` is the angular diameter of the sun as measured from earth and `D` is the distance to the sun, then the diameter of the sun `d_{sun}` is
`d_{sun}=2*D*tan (\theta/2)`.
Using the small angle approximation (`tan\theta~=\theta` in radians)
`d_{sun}= D*\theta` in `\theta` radians or
`d_{sun}= D*\pi/180*\theta` in `\theta` degrees.

Explanation:

Draw the sun, given the sun some size, draw a point to represent the location of the earth (this does NOT need to be to scale).
Draw a line from the location of the earth to the center of the sun.
Draw the diameter of the sun at right angles to this like.
Make an isosceles triangle by connecting the ends of the diameter to the loctaion of the earth. Should look something like this.

`\theta` the angular size of the sun is the angle bound by the diameter.

`\theta/2` is the little angle in the two right angle triangles.

`tan(\theta/2)=r_{sun}/D`

rearranging we have

`r_{sun}=D tan(\theta/2)`.

since `d_{sun}=2* r_{sun}`

`d_{sun}=2*D *tan(\theta/2)`.
Using the small angle approximation (which only works in radians) we have,
`d_{sun}=2*D* \theta/2=D *\theta_{radians}`.
If we have `\theta` in degrees we can convert using `\theta_{radians}=pi/180 \theta_{degrees}`
giving
`d_{sun}=pi/180 D *\theta_{degrees}`

note that `\theta_{degrees}` is around half a degree.