Standard Form of Circles

Algebra 2 · Axiom Academy

LESSON Standard Form of Circles Every point the same distance from a center — run through the distance formula, that one idea becomes an equation you can read like a map. 1. One Distance, Every Direction Pick any point (x,y) on a circle centered at (h,k) . The horizontal gap to the center is x-h and the vertical gap is y-k — the two legs of a right triangle whose hypotenuse is the radius . Watch the point travel around the circle: the legs stretch and shrink, but the hypotenuse never changes. That fixed hypotenuse is the whole definition of a circle. The distance from the point to the center equals r Square both sides — the standard form of a circle 2. Reading the Center and Radius Because (x-h)^2+(y-k)^2=r^2 is just the origin circle x^2+y^2=r^2 slid over to (h,k) , you can read a circle straight from its equation. The subtraction is the catch: x-h means the center sits at +h . Watch a radius-3 circle translate right 2 and down 1. Match each piece to the pattern (x-h)^2+(y-k)^2=r^2 : the x-2 gives h=2 , and y+1=y-(-1) gives k=-1 , so the center is (2,-1) . The right side is 9=3^2 , so r=3 . 3. Build It, Graph It, Check It Going the other way is just as direct. Given a center (-3,2) and radius 5 , drop the numbers into the pattern and simplify. Then graph it — plant the center, swing the radius all the way around — and confirm a point really lies on the circle by checking it satisfies the equation.

This is the written version of the interactive lesson above. See the full Algebra 2 course.