How do you find the center and radius of a circle?

Reading the center and radius from standard form

The standard equation of a circle is (x - h)^2 + (y - k)^2 = r^2. This formula comes directly from the Pythagorean theorem: every point (x, y) on the circle sits at a fixed distance, r, from the center point (h, k). When an equation is already in this form, you can find the center and radius of a circle by inspecting the numbers.

Pay close attention to the signs inside the parentheses. Because the standard equation has minus signs, (x - 3)^2 + (y + 5)^2 = 16 means h = 3 and k = -5. The center is (3, -5). The right-hand side represents r^2, so take the positive square root of 16 to find the radius, r = 4.

Using completing the square when the equation is expanded

Often, circle equations appear in general form, such as x^2 + y^2 - 6x + 8y - 11 = 0. To find the center and radius, rewrite the equation into standard form by completing the square for both x and y.

First, move the constant term to the right side: x^2 - 6x + y^2 + 8y = 11. Next, take half of each linear coefficient and square it. Half of -6 is -3, and (-3)^2 is 9. Half of 8 is 4, and 4^2 is 16. Add both values to both sides of the equation: (x^2 - 6x + 9) + (y^2 + 8y + 16) = 11 + 9 + 16.

Now condense each bracket into a squared binomial: (x - 3)^2 + (y + 4)^2 = 36. From this standard form, the center is (3, -4) and the radius is the square root of 36, which is 6.

Deciding if a point lies inside, on, or outside the circle

Once you know the equation (x - h)^2 + (y - k)^2 = r^2, you can test any point (a, b) by substituting a for x and b for y. Calculate the resulting value on the left side and compare it to r^2.

If the result equals r^2, the point lies exactly on the circle. If the result is less than r^2, the point is inside the boundary. If the result is greater than r^2, the point is outside the circle.