Calculate the radius of a circle or sphere given a section?

Question

This is probably a basic Math101 problem for most of you, but I'm not a mathematician so I could use some help with it.
In the diagram below, how would I solve for R (the radius of the circle or sphere), given A and B?

Answer 1

Consider drawing a line from the center of the circle to where the lower line intersects the circle (it does not matter whether it intersects in on the left or the right). This line is also a radius of the circle and forms a right triangle. Thus, by the Pythagorean Theorem, we can relate all of the sides together, but it requires that we know the length of the height of the triangle, which we are not explicitly given. However, the length from the center of the circle to the top (your red line) is $R$, and the length between the lines is $A$, so the height of the triangle is $R-A$. The width of the triangle is half the distance you provided $B/2$, so we have the relationship

$$R^2=\left(\frac{B}{2}\right)^2+(R-A)^2$$

From here, you can solve for $R$.