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What Is the Equation of a Circle? (Mathematics Lesson)

What Is the Equation of a Circle?

The equation of a circle is:



where (a, b) are the Cartesian coordinates of the center of the circle and r is the radius of the circle.

A Real Example of an Equation of a Circle




This is the equation of the circle centered on (2,3) with a radius of 5 (=√25).

Another Real Example of an Equation of a Circle




This is the equation of the circle centered on (-1,1) with a radius of 3 (=√9).

How to Find the Center and Radius of a Circle from its Equation

The center and radius of a circle can be found by looking at the equation of a circle.

Center



The x-coordinate is -1.



The y-coordinate is 1.

The center of the circle is (-1,1).

Radius



The radius is 3.

Read more about how to find the radius of a circle from the equation

Yet Another Example of the Equation of a Circle

The slider below shows another real example of the equation of a circle:

The General Form of an Equation of a Circle

The equation of a circle discussed above is the standard form.

If the squared brackets are expanded, the general form of the equation of a circle is found:



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Note
WHAT IS A CIRCLE?

A circle is a shape containing a set of points that are all the same distance from a given point, its center.



PARTS OF A CIRCLE

The center is the point the same distance from the points on the circle.

The radius is the line segment from the center of the circle to any point on the circle.



CIRCLE CENTERED AT THE ORIGIN

A circle centered at the origin has a centre at (0,0).

If it has a radius r, the equation is:

(x - 0)2 + (y - 0)2 = r2

x2 + y2 = r2



Top Tip

FINDING THE CENTER

The equation of a circle is:



The center is (a,b).

The number in the bracket with the x determines the x-coordinate.

The number in the bracket with the y determines the y-coordinate.

The a and b are being subtracted from the x and y.

This means numbers that are

  • negative in the brackets lead to a positive center co-ordinate

  • positive in the brackets lead to a negative center co-ordinate.


FINDING THE CENTER

Don't be confused if you see an equation which looks like this:

(x - 1)2 + (y - 3)2 - 49 = 0

This is still an equation of a circle:

(x - a)2 + (y - b)2 - r2 = 0