Equation of a Circle Formula
The circle is defined as the set of points placed at equal distance from a single point in a plane. The midpoint of the circle is called the center of the circle. In this section, we will learn the equation of a circle formula when the coordinates of the center and the radius are given. The equation of a circle formula is used for calculating the equation of a circle. Let us learn the equation of a circle formula with its derivation using the distance formula and a few solved examples.
What is the Equation of a Circle Formula?
\((x_1, y_1)\) is the center of the circle with radius r. (x, y) is an arbitrary point on the circumference of the circle. The distance between this point and the center is equal to the radius of the circle. So, let's apply the distance formula between these points.
\( \sqrt{(x  x_1)^2 + (y  y_1)^2} = r\)
Squaring both sides, we get: \((x  x_1)^2 + (y  y_1)^2 = r^2\). So, the equation of a circle is given by:
\((x  x_1)^2 + (y  y_1)^2 = r^2\)
You can use the equation of a circle calculator that helps to calculate the equation of a circle. Look at a few solved examples to understand the equation of a circle formula better.
Solved Examples Using Equation of a Circle Formula

Example 1: Find the center and radius of the circle whose equation is (x  1)^{2 }+ (y + 2)^{2} = 9.
Solution:
We will use the equation of a circle formula to determine the center and radius of the circle.Comparing \((x  1)^2 + (y + 2)^2 = 9\) with \((x  x_1)^2 + (y  y_1)^2 = r^2\), we get
\(x_1\) = 1, \(y_1\) = 2 and r = 3
So, the center and radius are (1, 2) and 3 respectively.
Answer: The center of the circle is (1, 2) and its radius is 3.

Example 2: What will be the equation of a circle if its center is at the origin?
Solution:
The equation of a circle is given by \((x  x_1)^2 + (y  y_1)^2 = r^2\).
If center is at origin, then \(x_1\) = 0 and \(y_1\) = 0.
Answer: The equation of the circle if its center is at origin is x^{2} + y^{2} = r^{2}.