Determining the Radius of a Circle
Given: Three Points on the Circumference
 Equation 1: (0 - h) ² + (21 - k) ² = r ² h ² + 441 - 42k + k ² = r ² 441 - 42k = r ² - h ² - k ² Equation 2: (13.5 - h) ² + (13.5 - k) ² = r ² 182.25 - 27h + h ² + 182.25 - 27k + k ² = r ² 364.5 - 27h - 27k = r ² - h ² - k ² Equation 3: (22 - h) ² + (0 - k) ² = r ² 484 - 44h + h ² + k ² = r ² 484 - 44h = r ² - h ² - k ² The right sides of the equations are equal, therefore all the equations are equal to one another. The following examples could have equated Equation 2 and Equation 3, and solved for h first rather than k. Left Side of Equation 1 = Left Side of Equation 2 441 - 42k = 364.5 - 27h - 27k -15k = -76.5 - 27h> k = (-76.5 - 27h) / -15 Left Side of Equation 1 = Left Side of Equation 3 441 - 42k = 484 - 44h - 42k = 43 - 44h k = (43 - 44h) / - 42 Both equations equal k, therefore the equations are equal to each other. (-76.5 - 27h) / -15 = (43 - 44h) / - 42 - 42(-76.5 - 27h) = -15(43 -- 44h) 3213 + 1134h = - 645 + 660h 474h = - 3858 h = - 8.13924 Substituting the value of h in either equation expressed in terms of k: k = (-76.5 - 27 × - 8.13924) / -15 = - 9.55063 or k = (43 - 44 × - 8.13924) / - 42 = - 9.55063 h and k may be substituted in any of the original equations to determine the radius of the circle. Equation 1: (0 - (- 8.13924)) ² + (21 - (- 9.55063)) ² = r ² 8.13924 ² + 30.55063 ² = r ² 66.24723 + 933.34099 = r ² 999.58822 = r ² Taking the square root of both sides of the equation: r = 31.61627 Equation 2: (13.5 - (- 8.13924)) ² + (13.5 - (- 9.55063)) ² = r ² 21.63924 ² + 23.05063 ² = r ² 468.25671 + 531.33154 = r ² 999.58825 = r ² Taking the square root of both sides of the equation: r = 31.61627 Equation 3: (22 - (- 8.13924)) ² + (0 - (- 9.55063)) ² = r ² 30.13924 ² + 9.55603 ² = r ² 908.37379 + 91.21453 = r ² 999.58832 = r ² Taking the square root of both sides of the equation: r = 31.61627