This is a classic problem that goes back to the Babylonians and is called the "bow and arrow" problem. (The arc of the circle and chord are the bow and bowstring. The radius through the center is the arrow.)
Call the length of the chord "l" and the height, from chord to arc at the center of the arc, "h". Call the (unknown) radius of the circle "r". The distance from the center of the circle to either end of the arc is r, the radius. The distance from the center of the circle to the center of the chord is r- h. The distance from the center of the chord to one endpoint is l/2. Those lines form a right triangle so, by the Pythagorean theorm, which, multiplying that first square, is and then, canceling the " " terms, reduces to . We can write that as and so .