Center of Circle
A circle is defined as the locus of a moving point on a plane such that its distance from a fixed point on the plane remains constant or fixed. That fixed point is called the center of the circle. Let us learn more about the center of a circle in this article.
1.  Center of Circle Definition 
2.  Center of Circle Formula 
3.  How to Find Center of Circle? 
4.  FAQs on Center of Circle 
Center of Circle Definition
A circle is a 2D shape defined by its center and radius. We can draw any circle if we know the center of circle and its radius. A circle can have an infinite number of radius. The center of a circle is the midpoint where all the radii meet. It can also be defined as the midpoint of the diameter of the circle. Look at the diagram given below where O is the center of circle and OP is the radius.
Center of Circle Formula
In a circle, if the coordinates of the center are (h,k), r is the radius, and (x,y) is any point on the circle, then the center of circle formula is given below:
(x  h)^{2} + (y  k)^{2} = r^{2}
This is also known as the center of the circle equation. We will be using this formula in the sections below to find the center of a circle or the equation of the circle.
How to Find Center of Circle?
Now, we will learn how to find the center of the circle using some simple steps. There are two cases that might come up when you would be asked to find the center of a circle:
 When a circle is given, and we have to find its center.
 When an equation of a circle is given, and we have to find the coordinates of its center.
When a Circle is Given
When a circle is given to us and we have to find its center point, then we can follow the steps listed below:
Step 1: Draw a chord PQ in a circle and carefully note its length (which is 4 inches in the figure below).
Step 2: Draw another chord MN parallel to PQ such that it should be of the same length as PQ.
Step 3: Join the points P and N through a line segment using a ruler.
Step 4: Join points Q and M.
Step 5: The point of intersection of PN and QM is the center of the circle.
When Equation of the Circle is Given
Till now, you are familiar with the center of circle equation (given above). Now, if you are given an equation of a circle, and you will have to find its center, then how will you do it? Suppose we have to find the coordinates of the center of a circle with equation x^{2} + y^{2}  4x  6y  87 = 0, then the steps to do the same are listed below:
 Step 1: Write the given equation in the form of the general equation of a circle  (x  h)^{2} + (y  k)^{2} = r^{2}, by adding or subtracting numbers on both sides.
We can write the given equation as x^{2}  4x + y^{2}  6y = 87. Add 4 to both sides of the equation to get a perfect square of x2. So, we will get, x^{2}  4x + 4 + y^{2}  6y = 87 + 4.
⇒ (x  2)^{2} + y^{2}  6y = 91
Add 9 to both sides to get a perfect square of y3.
⇒ (x  2)^{2} + y^{2}  6y + 9 = 91 + 9
⇒ (x  2)^{2} + (y  3)^{2} = 100
⇒ (x  2)^{2} + (y  3)^{2} = 10^{2}
This looks like the general equation of circle.
 Step 2: Compare this equation with the general equation and identify the values of h, k, and r.
If we compare (x  2)^{2} + (y  3)^{2} = 10^{2} with (x  h)^{2} + (y  k)^{2} = r^{2}, we can identify that h = 2, k = 3, and r = 10. So, we have got the coordinates of the center of circle which are (h, k) = (2, 3).
How to Find Center of Circle with Two Points?
If the endpoints of the diameter of the circle are given, then to find the coordinates of the center we have to use the midpoint formula, as the center is the midpoint of the diameter of the circle. The steps to find the center of a circle with two points are given below:
 Step 1: Assume that the coordinates of the center of the circle are (h, k).
 Step 2: Use the midpoint formula which says that if (h, k) are the coordinates of the midpoint of a segment with endpoints (x_{1}, y_{1}) and (x_{2}, y_{2}), then (h, k) = (x_{1} + x_{2}/2, y_{1} + y_{2} /2).
 Step 3: Simplify it and get the coordinates of the center of the circle.
Let us take an example of a circle in which the endpoints of a diameter are given as (2, 4), and (6, 16). Then, the coordinates of its center are:
(h, k) = (2+6/2, 4+16/2)
(h, k) = (4/2, 20/2)
(h, k) = (2, 10)
Therefore, the coordinates of the center of a circle with the endpoints of diameter are (2, 10).
Related Articles on Center of Circle
Check these interesting articles related to the concept of center of circle in geometry.
Center of Circle Examples

Example 1: Find the equation of the center of a circle if the coordinates of the center are (0, 0) and the radius of the circle is 5 units.
Solution: The center of the circle equation is (x  h)^{2} + (y  k)^{2} = r^{2}. The given values are: coordinates of the center (h, k) are (0, 0), and the radius (r) = 5 units. Substituting the values of h, k, and r in the equation, we get, (x  0)^{2} + (y  0)^{2} = 5^{2}. After simplifying it, we get, x^{2} + y^{2} = 25. This is the required equation of center of circle or simply the equation of the circle.

Example 2: What will be the coordinates of the center of the circle, if the endpoints of the diameter are (8, 7) and (4, 5)?
Solution: We know that the center is the midpoint of the diameter of a circle. If the coordinates of the endpoints of the diameter are (8, 7) and (4, 5), then the coordinates of the center of the circle are:
(h, k) = ((8 + 4)/2, (7 + 5)/2)
= (12/2, 2/2)
= (6, 1)
Therefore, the coordinates of the center are (6, 1).
FAQs on Center of Circle
What is the Radius and Center of Circle?
The center of a circle is the center point of the circle. It is the point where we place the tip of our compass while drawing a circle. It is the midpoint of the diameter of the circle. In a circle, the distance between the center to any point on the circumference is always the same which is called the radius of the circle. It is half of the length of the diameter.
What are the Coordinates for the Center of the Circle and the Length of the Radius?
The coordinates of the center of the circle are the distance of the center point from the xaxis and yaxis respectively. It is generally denoted in the form of (h, k), where h and k represent x and y coordinates respectively. The length of the radius is denoted by r. The coordinates of the center and the radius are related to each other in the form of an equation: (x  h)^{2} + (y  k)^{2} = r^{2}.
What is the Center of a Circle Represented by the Equation (x−5)^{2}+(y+6)^{2}=42?
If we compare the given equation with the general equation of center of circle  (x  h)^{2} + (y  k)^{2} = r^{2}, we can see that h = 5, k = 6, and r = √42. So, the center of the circle is at (5, 6).
How to Find Center of Circle?
To find the center of a circle, we can draw two parallel chords having the same length inside the circle. Then, join the opposite ends of the chords. That point of intersection will be the center of the circle.
How to Find Center of Circle with Endpoints of Diameter?
The center of a circle is the midpoint of the diameter. So, by using the midpoint formula, if the endpoints of the diameter are (a, b) and (c, d), then the coordinates of the center of circle are ((a + c)/2, (b + d)/2).
How to Find Radius and Center of Circle from Equation?
If the equation of a circle is given, then we can find its radius and center by comparing it with the general form of the equation  (x  h)^{2} + (y  k)^{2} = r^{2}. We will find the values of h, k, and r. Then, (h, k) will be the coordinates of the center of circle and r will be the radius.
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