Circle Equation Calculator — Center & Radius

Use this free circle equation calculator to turn the general form x² + y² + Dx + Ey + F = 0 into center-radius form. It returns the center, radius, circumference, area, and a graph.

How it works

Completing the square gives center (−D/2, −E/2) and radius r = √[(D/2)² + (E/2)² − F].

How to use it

1. Enter D, E, and F.
2. Press Find center & radius.

Related: the Circle Calculator and more free math calculators.

Worked Example: General Form to Center and Radius

Find the center and radius of x² + y² − 6x + 4y − 12 = 0.

1. Group the x terms and the y terms, and move the constant: (x² − 6x) + (y² + 4y) = 12.
2. Complete the square for x. Half of −6 is −3, and (−3)² = 9. Add 9 to both sides.
3. Complete the square for y. Half of 4 is 2, and 2² = 4. Add 4 to both sides.
4. Rewrite: (x − 3)² + (y + 2)² = 12 + 9 + 4 = 25.
5. Read the answer: center (3, −2), radius √25 = 5.

Check with the shortcut: D = −6, E = 4, F = −12. The center is (6/2, −4/2) = (3, −2), and r² = 9 + 4 + 12 = 25. From there, the circumference is 2π(5) = 10π ≈ 31.42, and the area is π(5²) = 25π ≈ 78.54.

Writing a Circle Equation from Other Information

Sometimes you know the center and a point instead of the equation. Here is how to build the equation by hand.

From the center and radius

Center (−2, 5) and radius 4: substitute h = −2, k = 5 and r = 4 to get (x + 2)² + (y − 5)² = 16. Watch the signs. Subtracting a negative h turns into a plus.

From the center and a point on the circle

Center (1, −4) and point (4, 0): the radius is the distance between them. r² = (4 − 1)² + (0 − (−4))² = 9 + 16 = 25, so the equation is (x − 1)² + (y + 4)² = 25.

From the endpoints of a diameter

Endpoints (−1, 2) and (5, 10): the center is the midpoint, (2, 6). The radius is half the diameter. The diameter is √(6² + 8²) = 10, so r = 5 and the equation is (x − 2)² + (y − 6)² = 25.

Frequently asked questions

What is the center-radius form of the equation of a circle?

The center-radius form, also called standard form, is (x − h)² + (y − k)² = r², where (h, k) is the center and r is the radius. For example, a circle with center (−1, 4) and radius 3 is (x + 1)² + (y − 4)² = 9. Notice that subtracting a negative coordinate turns into a plus sign.

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

Complete the square, or use the shortcut for x² + y² + Dx + Ey + F = 0: the center is (−D/2, −E/2) and r = √((D/2)² + (E/2)² − F). For x² + y² + 8x − 2y + 8 = 0, the center is (−4, 1) and r² = 16 + 1 − 8 = 9, so the radius is 3.

How do you find the equation of a circle?

You need the center and the radius. If you know the center and one point on the circle, use the distance formula to get r², then write (x − h)² + (y − k)² = r². For center (2, 3) through (5, 7), r² = 3² + 4² = 25, so the equation is (x − 2)² + (y − 3)² = 25.

How do you graph a circle from its center and radius?

Plot the center, then count the radius straight up, down, left and right to mark four points, and connect them with a smooth round curve. For (x − 2)² + (y + 1)² = 16, the center is (2, −1) and the radius is 4, giving the points (6, −1), (−2, −1), (2, 3) and (2, −5).

How do you put a circle equation in a graphing calculator?

Desmos accepts a circle equation exactly as written. On a TI-84, solve for y and enter two functions, one for the top half and one for the bottom half. For (x − 1)² + y² = 9, enter Y1 = √(9 − (X − 1)²) and Y2 = −√(9 − (X − 1)²), then use a square zoom so the circle does not look stretched.

How to use the Circle Equation Calculator for homework

The Circle Equation Calculator is most useful when you treat it as a learning check, not just a shortcut to the final answer. Start by copying the original problem carefully, including signs, exponents, decimal points, fractions, parentheses, and units. Then enter the values in the same order the problem gives them. A small typing change can completely change the result, especially in algebra, statistics, geometry, and probability problems.

Before you press the button to calculate, make a quick estimate or prediction. The estimate does not need to be exact. Its job is to help you notice impossible answers. If a distance becomes negative, a probability is bigger than 1, an angle looks too large, or a decimal point seems misplaced, go back and check the input before trusting the final result.

Before you enter the problem

• Rewrite the problem in a clean line so every value is easy to see.
• Use parentheses around grouped expressions, especially in fractions and exponents.
• Keep units with the numbers while you work, even if the calculator only asks for the numbers.
• Check whether the problem wants an exact value, a decimal approximation, or a rounded answer.
• Look for restrictions such as positive values only, a chosen interval, or a required domain.

How to read the result

After the calculator gives a result, read more than the final line. If steps, tables, graphs, or intermediate values are shown, use them to understand how the answer was built. That is especially important when you are studying for a quiz or test, because teachers often give more credit for a correct process than for an unsupported number.

Try to identify the main idea behind the result. For example, ask yourself which formula was used, which operation changed the expression, which value controlled the graph, or which assumption made the answer possible. When you can explain that idea in your own words, the tool has helped you learn the skill instead of only checking one problem.

Common mistakes to avoid

Turn the answer into practice

One good way to study is to solve the problem by hand first, then use the Circle Equation Calculator to check your work. If your answer is wrong, do not erase everything immediately. Find the first line where your work stops matching the calculator’s logic. That line is usually where the real misunderstanding happened.

1. Work the problem on paper and circle your final answer.
2. Use the calculator to check the result.
3. If the answers match, write one sentence explaining the method.
4. If they do not match, compare each step until you find the first difference.
5. Redo a similar problem without the tool to make sure the correction sticks.

When to use a calculator and when to work by hand

Use the Circle Equation Calculator when you want to check a long calculation, explore a pattern, test a graph, or confirm a result after practicing. Work by hand when the assignment asks for steps, when you are learning a new method, or when the test will not allow a digital tool. The strongest students use both: hand work to build understanding and calculators to check accuracy.

If you are preparing for a timed test, practice some problems without the tool and some with it. That balance helps you build speed while still understanding the math. Over time, the goal is to need the calculator less often for routine steps and use it more strategically for checking, exploring, and confirming your reasoning.

Quick accuracy checklist

Before you leave the page, use the Circle Equation Calculator result to answer three quick questions: What was the input? What rule, formula, or method connects the input to the result? What would change if one number in the problem changed? These questions turn a single calculator answer into a reusable math skill.

For homework, write the original problem, the calculator’s result, and one short explanation of the method in your notebook. For test review, cover the result and try to reproduce the setup on your own. If you can set up a similar problem without looking back at the calculator, you are much closer to owning the skill.