Equations of Circles

ACT Math · Axiom Academy

Every circle equation is really just a distance statement — center, radius, and one squared-distance formula unlock every ACT circle question. 1. Standard Form — Reading Off Center and Radius Standard form packages the distance definition directly: (x-h)^2+(y-k)^2=r^2 , where (h,k) is the center and r is the radius. Watch the animation — the point sliding around the edge is always exactly r away from the center, no matter where it sits. That's not a coincidence; it's the definition of a circle, made visible. Every point on the circle is distance r from (h, k) Find the center and radius of (x-2)^2+(y+3)^2=25 . Center: (2,-3) . Radius: 5 (since r^2=25 ). 2. Building the Equation — From a Center, or From a Diameter The ACT often runs standard form in reverse: it gives you the center and radius (or two other pieces of information) and asks for the equation. Plug directly into (x-h)^2+(y-k)^2=r^2 . When you're given a diameter's endpoints instead: the ACT sometimes hands you two points on the circle that happen to be endpoints of a diameter, not the center and radius directly. Three steps get you there: Find the equation of the circle with diameter endpoints A(1,2) and B(7,8) . Center (4,5) , radius , so r^2=18 : (x-4)^2+(y-5)^2=18 . 3. Completing the Square — Unscrambling General Form

This is the written version of the interactive lesson above. See the full ACT Math course.