Suppose we try to find the circumradius of a triangle , as shown below:

How can we find it? Note that the triangle is isosceles, with the equal sides being radii of the circumcircle. To find these, we only need to know the angle and side length AC. Angle intercepts the arc , with O at the center of the circle. Angle also intercepts the same arc, but with point B on the circumference of the circle. This means

(1)  

where symbol ‘m’ denotes magnitude of an angle. We are now ready to solve it. Construct point D as a mid-point of segment . Segment is an angle bisector, meaning

(2)  

Therefore, =

Note that is a right-angled triangle with hypotenuse OC. The length of segment can be found by noting that:

(3)  

Solving for OC, we find that

However, since , this can be written as

, or , where R is the circumradius

An important discovery here is that we could have done this for any side and its opposite angle and we should have gotten the same result for the circumradius. This means that

(4)  

This is an important relationship, known as Law of Sines, that holds true for any triangle.