Center of Circle (original) (raw)

Last Updated : 15 Apr, 2026

Center is defined as a point inside the circle that is equidistant from all the points on the circumference.

diametrical

Center of Circle Formula

For any general point (x, y) on the circle with radius r, the coordinates of the center (h, k) are given by the following formula:

(x-h)2 + (y-k)2 = r2

Ways to Find the Centre of a Circle

Only the Circle is given:

**Step 1: Draw a chord AB in the circle and measure its length.

Center-of-Circle-Formula-1

**Step 2: Draw another chord, CD, parallel to AB, with the same length as AB.

Center-of-Circle-Formula-2

**Step 3: Use a ruler to connect points C and B with a straight line segment.

Center-of-Circle-Formula-3

**Step 4: Similarly, connect points A and D with a straight line.

Center-of-Circle-Formula-4

The point where the two lines AD and BC intersect will be the center of the circle.

Two Points are given:

If the endpoints of the Diameter of Circle are given, the center is by definition the midpoint of the line segment connecting them. To find the center:

**Step 1: Let the two given points be A(x1, y1​) and B(x2, y2​).

**Step 2: The coordinates of the center (h, k) are the average of the x-coordinates and the average of the y-coordinates.

**Step 3: Use Midpoint formula : (h, k) = [(x1 + x2) / 2, (y1 + y2) / 2].

**Step 4: The midpoint (h, k) is the center of the circle.

2056957788

**Example: Find the center of a circle, for instance, that passes through (3, 4) and (-3, -4) that are on the opposite endpoints of the diameter of the circle.

**Solution:

Let center of circle is (h, k)

(h, k) is the mid-point of the two end-points on of the diameter
(h, k) = [(-3 + 3) / 2, (4 - 4) / 2]
(h, k) = (0, 0)

(0, 0) represents the center of this circle.

Equation of Circle is given:

When the circle's equation is expressed in the standard form (x - h) + (y - k) = r2, the center's coordinates are ****(h, k)**, and the radius is denoted by **r.

Expressing Center of Circle

If you are given an equation for a circle or just two points on it, there are multiple ways to determine its center. Some of the common ways are,

Using Chords

Line segments that join two points on a circle are called chords. A chord is referred to as a diameter if it runs through the center of the circle. The circle's center is the midpoint of a diameter. Given the endpoints of a chord (diameter), (x1, y1) and (x2, y2), the formula to find the midpoint of the chord is,

****((x** 1 + x 2 )/2, (y 1 + y 2 )/2) is the midpoint.

Using a Secant

A line that crosses a circle twice is called a secant of a circle. The line segment that results when a secant passes through the circle's center is called a diameter. The center of the circle is the midpoint of the diameter, just like in the chord case.

Using Tangents

The radius drawn to the point of tangency of a circle is perpendicular to the tangent line. Therefore, the center of the circle lies on the line passing through the point of tangency and perpendicular to the tangent.

When Equation of Circle is Given to Us

A circle's standard equation is ****(x - h) + (y - k) = r** 2, where the radius is denoted by r and the circle's center by (h, k). Therefore, you can determine the values of h and k, which stand for the center's coordinates, from the provided equation.

**Example: On the circle, given points A(3, 4) and B(7, 8) find its center.

**Solution:

Suppose we have **A(3, 4) and **B(7, 8), two points on the circle.

**Step 1: Determine the Circle's Two Points
Select two locations within the circle. In this instance, our selected points are A and B.

**Step 2: Use the Midpoint Calculation
To determine the midpoint's coordinates (h, k), use the following formula:
h = (x1 ​+ x2)​/2
k = (y1​ + y2)/2​​

Enter A and B's coordinates into the formula:
h = (3 + 7)/2 = 5
k = (4 + 8)/2 = 6

So, the midpoint (h, k) is (5, 6).

**Step 3: Center is represented by midpoint coordinates
Circle's center is represented by the coordinates (5, 6) that were found using the midpoint formula.

**Step 4: Double-check using the additional points (optional)
You can verify that (5,6) is equally spaced from any additional points on the circle if there are any available points.

**Step 5: Examine the Circle Equation (Alternative Method)
If the circle's equation is known, using the formula should also produce (5, 6) as the center.

Therefore, in this case, (5, 6) is the center of the circle formed by points A(3, 4) and B(7, 8).

Solved Examples

**Example 1: The task at hand is to determine the radius and center of the circle that is shown by the equation (x − 3) + (y + 2) = 25.

**Solution:

Given, ****(x − 3)** 2 **+ (y + 2) 2 **= 25

By comparing with the standard form ****(x − h)** 2 **+ (y − k) 2 **= r 2
We determine radius and the center is at,
r = 5
(h, k) = (3, −2)

**Example 2: Find the radius and center of the circle that passes through the three points A(1, 2), B(5, 6), and C(−3, 4).

**Solution:

Let the equation of the circle be x2+y2+Dx+Ey+F=0.
Since the circle passes through A(1,2), B(5,6) and C(−3,4) substitute each point :

For A (1, 2): 5 + D + 2E + F = 0. ....(1)

For B (5, 6): 61 + 5D + 6E + F = 0. ....(2)

For C (-3, 4): 25 − 3D + 4E + F = 0. ....(3)

Solving Equation (1) , (2) and (3), we get:

D = -4/3 , E = -38/3.

Thus the center is\left(-\frac{D}{2}, -\frac{E}{2}\right) = \left(\frac{2}{3}, \frac{19}{3}\right)

**Example 3: Given the Circle's Equation: (x − 3) + (y + 4) = 25.

**Solution:

Let us take the case where we have the circle equation ****(x − 3)** 2 **+ (y + 4) 2 **= 25.
First, determine center coordinates.

Equation allows us to determine center coordinates (h,k) directly:
h = 3
k = −4

Thus, (3, -4) is circle's center.

Practice Questions

**Question 1: Determine the circle's equation with radius 4 and center (−2, 3).

**Question 2: The circle represented by the equation x2 + y2 - 6x + 8y + 9 = 0 has a center and a radius.

**Question 3: Determine the equation of the circle with the midpoint of PQ at its center, given two points, P(2, 5) and Q(−3, −1).

**Question 4: If the circle x2 + y2 - 2x + 4y - 13 = 0, find the new center and radius of the translated circle after being translated three units to the right and two units upward.