Circle Equation in Standard Form: Step-by-Step Guide
A circle with its center at (h, k) and a radius of r has the equation (x - h)^2 + (y - k)^2 = r^2. If you know the center and radius, you can write the equation right away. If you are given a messier equation, you can complete the square on the x terms and the y terms to find the center and radius.
Math Racer skill guide · Geometry · Grade 10
How to solve it, step by step
- Find the center and the radius. Find the center (h, k) and the radius r from the problem. If you only have the messy form, go to step 4 first.\[\text{center} = (h, k),\; \text{radius} = r\]
- Put them into the formula. Replace h, k, and r in (x - h)^2 + (y - k)^2 = r^2.\[(x - h)^2 + (y - k)^2 = r^2\]
- Clean up the signs. If h or k is negative, the minus becomes a plus. For example, x - (-3) becomes x + 3.\[(x - (-h))^2 = (x + h)^2\]
- Complete the square (if you start from the messy form). Group the x terms and the y terms. Add the right number to each group to make perfect squares, and add the same numbers to the other side.\[x^2 + Dx + \left(\tfrac{D}{2}\right)^2 + y^2 + Ey + \left(\tfrac{E}{2}\right)^2 = -F + \left(\tfrac{D}{2}\right)^2 + \left(\tfrac{E}{2}\right)^2\]
- Check with a point. If you know a point on the circle, put it into the equation. It should work.\[(x_0 - h)^2 + (y_0 - k)^2 \stackrel{?}{=} r^2\]
Worked example
Write the equation of the circle with center (-3, 4) and radius 5 in standard form.
Step 1: The center is (h, k) = (-3, 4) and the radius is r = 5.
Step 2: Put them into the formula: (x - (-3))^2 + (y - 4)^2 = 5^2.
Step 3: Clean up the double negative and square the 5: (x + 3)^2 + (y - 4)^2 = 25.
Step 4: Check with a point. (2, 4) is 5 units to the right of the center, so it should be on the circle: (2 + 3)^2 + (4 - 4)^2 = 25 + 0 = 25. It works.
The equation is (x + 3)^2 + (y - 4)^2 = 25.
Tips and common mistakes
- Inside the parentheses you subtract the center: (x - h) and (y - k). If h is negative, you end up with a plus.
- The right side is r^2, not r. Square the radius before you write the equation.
- To read the center from an equation, flip the signs. (x + 3)^2 means h = -3.
- If you finish completing the square and r^2 is negative, there is no circle.
- If r^2 turns out to be 0, the graph is just one point.
- When you start from the messy form, make sure x^2 and y^2 each have a 1 in front. If not, divide the whole equation first.
- Helpful reminders: diameter = 2r, circumference = 2 x pi x r, and area = pi x r^2.