Center of Circle Calculator
Enter a circle's equation, three points on it or the ends of a diameter to find its center and radius.
Updated
Center of Circle Calculator
Our Center of Circle Calculator finds the center (h, k) and the radius of a circle from its equation in standard, general or parametric form, from three points on the circle, or from the two ends of a diameter. It also draws the circle and gives the equation in the other form, the diameter, circumference and area.
Quick answer
The circle (x − h)² + (y − k)² = r² has its center at (h, k): (x − 2)² + (y + 3)² = 25 has center (2, −3) and radius 5.
- General form: x² + y² + Dx + Ey + F = 0 has center (−D/2, −E/2)
- Three points: the perpendicular bisectors of two chords cross at the center
- Ends of a diameter: the center is their midpoint
- Watch the signs: (x + 3)² means h = −3
What Is the Center of a Circle?
The center is the one point that is the same distance from every point on the circle. That distance is the radius. On a graph the center is written as a coordinate pair (h, k), and together with the radius it fixes the circle completely.
Center From the Standard Form
center (h, k), radius r
Read h and k from inside the brackets, and change their signs: the formula subtracts them. The number on the right is r², so take its square root for the radius.
| Equation | Center, radius |
|---|---|
| (x − 2)² + (y + 3)² = 25 | (2, −3), 5 |
| x² + y² = 49 | (0, 0), 7 |
| (x + 1)² + y² = 20 | (−1, 0), 2√5 |
| (x − 4)² + (y − 4)² = 2.25 | (4, 4), 1.5 |
Center From the General Form
In x² + y² + Dx + Ey + F = 0 the center is (−D/2, −E/2) and r² = D²/4 + E²/4 − F. To see why, complete the square on x and on y:
(x² − 4x + 4) + (y² + 6y + 9)
= 12 + 4 + 9
(x − 2)² + (y + 3)² = 25
So the center is (2, −3) and the radius is 5. If D²/4 + E²/4 − F is 0 the "circle" is a single point, and if it is negative there is no real circle at all.
Center From Three Points
Any three points that are not on one straight line lie on exactly one circle. Join them with two chords and draw the perpendicular bisector of each (the line through the chord's midpoint at a right angle). The bisectors cross at the center.
Example: (2, 2), (7, −3) and (−3, −3) all lie on the circle with center (2, −3) and radius 5, since each one is 5 away from it.
Center From the Ends of a Diameter
The center sits halfway along every diameter, so it is the midpoint of the two ends. The radius is half the distance between them.
k = (y₁ + y₂) ÷ 2
(−1, 1) and (5, −7): (2, −3)
Center From the Parametric Form
y = k + r sin α
The constant terms are the center. As the angle α goes from 0° to 360° the point travels once around the circle. With center (2, −3), r = 5 and α = 60° the point is (4.5, 1.330127).
Finding the Center of a Real Circle
- Draw two chords. Any two straight lines across the circle, not parallel.
- Bisect each one. Mark its midpoint and draw a line at 90° through it.
- Mark the crossing. That point is the center.
For a paper circle, fold it in half twice in different directions: the two creases are diameters and cross at the center. To get the size, use the Diameter of Circle Calculator or the Area of Circle Calculator.
Common Mistakes
- Keeping the sign. (x + 3)² gives h = −3, not 3.
- Using r² as the radius. In (x − 2)² + (y + 3)² = 25 the radius is 5, not 25.
- Halving without the minus. From the general form h = −D/2, so D = −4 gives h = 2.
- Leading coefficients. Divide 2x² + 2y² − 8x + 12y − 24 = 0 by 2 before reading D, E and F.
Key Terms
- Center
- The point (h, k) the same distance from every point on the circle.
- Radius
- That distance, r.
- Chord
- A straight line joining two points on the circle.
- Bisector
- The line at 90° through a chord's midpoint. It passes through the center.
How to Use the Center of Circle Calculator
- Choose what you know. Pick Standard form, General form, Parametric, 3 points or Diameter ends. For the general form, first divide the equation so x² and y² have a coefficient of 1.
- Enter the numbers. Type the coefficients or the coordinates, such as A = 2, B = -3 and C = 25. Negative numbers, decimals and fractions such as 3/4 work, and pasting (2, -3) into an x field fills both boxes.
- Read and copy the center. The center (h, k) appears as you type, with the radius, a drawing on axes, the equation in the other form, and the diameter, circumference and area. Press Copy to copy it, or Clear or Esc to start again.
Example: for x² + y² − 4x + 6y − 12 = 0, choose General form and enter D = −4, E = 6 and F = −12. The center is (2, −3), the radius is 5 and the standard form is (x − 2)² + (y + 3)² = 25.
Who Can Use This Calculator?
Frequently Asked Questions (FAQs)
How do you find the center of a circle from its equation?
Write it in standard form (x − h)² + (y − k)² = r². The center is (h, k), with the signs changed from those inside the brackets.
What is the center of (x − 2)² + (y + 3)² = 25?
The center is (2, −3) and the radius is √25 = 5.
How do you find the center from the general form?
For x² + y² + Dx + Ey + F = 0 the center is (−D/2, −E/2). For x² + y² − 4x + 6y − 12 = 0 that is (2, −3), and r² = 4 + 9 + 12 = 25.
How do you find the center of a circle from three points?
Draw two chords between the points and the perpendicular bisector of each. They cross at the center. The points (2, 2), (7, −3) and (−3, −3) give the center (2, −3).
How do you find the center from the endpoints of a diameter?
Take the midpoint: average the x values and the y values. The ends (−1, 1) and (5, −7) give the center (2, −3).
How do you find the center of a circle with a ruler?
Draw two chords, mark the midpoint of each, and draw a line at 90° through each midpoint. The lines cross at the center.
Why does (x + 3)² mean the center is at x = −3?
The standard form subtracts h, so x + 3 is x − (−3) and h = −3.
When is x² + y² + Dx + Ey + F = 0 not a circle?
When D²/4 + E²/4 − F is 0 or less. At 0 it is a single point; below 0 there is no real circle.
Is the Center of Circle Calculator free?
Yes. It is completely free, needs no sign-up, and can be used as many times as you like.
Try our Center of Circle Calculator to find the center and radius of any circle from its equation, three points or a diameter.