Length of a line between origin and edge of a circle

[jack476]

For a circle with radius R centered at R along the X-axis so that the edge of the circle touches the origin, what is the length of a line drawn between the origin and an edge of the circle in terms of the angle between that line and the X-axis? This isn't a homework problem, just something I'm trying to figure out.

IE in this picture (sorry about the MS Paint, I'm on a fairly old computer right now), what is the length of the line L in terms of the angle a and radius R? Any help is much appreciated, thank you.

 

[fresh_42]

Are you firm in computing sides of triangles?

 

[jack476]

Are you firm in computing sides of triangles?

Yes, I'm pretty comfortable with that. What are you suggesting?

 

[fresh_42]

There are 2 possible triangles you can use to compute L depending on a. One involves Thales' Theorem. But it also can be done without. Just look for triangles and right angles you can find in your graphic.

 

[SteamKing]

For a circle with radius R centered at R along the X-axis so that the edge of the circle touches the origin, what is the length of a line drawn between the origin and an edge of the circle in terms of the angle between that line and the X-axis? This isn't a homework problem, just something I'm trying to figure out.

IE in this picture (sorry about the MS Paint, I'm on a fairly old computer right now), what is the length of the line L in terms of the angle a and radius R? Any help is much appreciated, thank you.

This article shows the trigonometry of working out L based on a circle with unit radius:

http://geowords.com/e_/06_wind&compass/pi/protractor.htm

Scroll down to 2) at the link. The diagram there:

​

Matches what you drew in the OP.

Taking the diameter of an arbitrary circle as D, which is the length of EB in the picture above, then cos α = AE / EB, or using your diagram, cos α = L / D.

Therefore, L = D cos α

or L = 2R cos α