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 radii. 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. Observe the figure given below where O is the center of circle and OP is the radius.
Center of Circle Formula
The center of circle formula is also known as the general equation of a circle. 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 following sections to find the center of a circle or the equation of the circle.
How to Find the Center of Circle?
In order to find the center of the circle, we will use some simple steps. There are two cases that might come up when we could be asked to find the center of a circle:
 When a circle is given and we need to find its center.
 When an equation of a circle is given and we need to find the coordinates of its center.
When a Circle is Given
When a circle is given to us and we need 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
If we know the equation of a circle, and we need to find its center, then we will use the following steps. Let us understand this with the help of an example.
Example: Let us find the coordinates of the center of a circle with equation x^{2} + y^{2}  4x  6y  87 = 0
Solution: The steps to find the coordinates of the center of a circle 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 y  3
⇒ (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 the 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 use the midpoint formula, because 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).
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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 can be calculated using the midpoint formula: (h, k) = [(x_{1} + x_{2}]/2, [y_{1} + y_{2}]/2).
After substituting the value of x_{1} = 8, x_{2} = 4, y_{1} = 7 y_{2} = 5
(h, k) = [(8 + 4)/2, (7 + 5)/2]
= (12/2, 2/2)
= (6, 1)
Therefore, the coordinates of the center are (6, 1).

Example 3: State true or false.
a.) 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 expressed as: (x  h)^{2} + (y  k)^{2} = r^{2}
b.) If the endpoints of the diameter of the circle are given, then to find the coordinates of the center we use the midpoint formula,
Solution:
a.)True, 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 expressed as: (x  h)^{2} + (y  k)^{2} = r^{2}
b.) True, if the endpoints of the diameter of the circle are given, then to find the coordinates of the center we use the midpoint formula.
FAQs on Center of Circle
What is the Center of Circle?
The center of a circle 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.
What are the Coordinates for the Center of the Circle and the Length of the Radius?
The coordinates of the center of the circle represent 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 the 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. The circle is also part of a conic section and the foci of the conic is 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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