GeoProof — Law of Sines & the Circumradius

a / sin A = b / sin B = c / sin C = 2R

Law of Sines & the Circumradius

The law of sines says the ratio of a side to the sine of its opposite angle is the same for all three sides. The extended form names that common value — it's the diameter of the circumscribed circle:

a/sin A = b/sin B = c/sin C = 2R

R is the circumradius: the radius of the one circle passing through all three vertices.

Why 2R

Fix side a = BC and draw the diameter from B. The triangle it forms with C has a right angle at C (an angle in a semicircle), and its angle at the far end equals A (both stand on arc BC). So sin A = a / (2R), i.e. a/sin A = 2R.

What it's good for

It finds the circumradius straight from one side and its opposite angle, links to the area formula Area = abc/(4R), and explains why a fixed side subtends the same angle from anywhere on its arc.