Equation of a Circle (Not Centred at the Origin)

The equation of a circle, with a centre with Cartesian coordinates (a, b) is in the form:

In this equation,

• x and y are the Cartesian coordinates of points on the (boundary of the) circle.
• a and b are the Cartesian coordinates of the centre of the circle.
• r is the radius of the circle.

The image below shows what we mean by a point on a circle centred at (a, b) and its radius:

Real Examples of Equations of Circles (Not Centred on the Origin)

It is easier to understand the equation of a circle with examples.

• A circle centred at (2, 3) with a radius of 5 will have the equation:
  
  
• A circle centred at (−1, 1) with a radius of 3 will have the equation:
  
  

Beware

The Center of a Circle Has Negative Coordinates

The equation of a circle is:

The center is (a, b).

• The number being subtracted from the x in the brackets is the x-coordinate of the center.
• The number being subtracted from the y in the brackets is the y-coordinate of the center.

What if a coordinate of the center is negative? Imagine the center of the circle is (−1, 2). The equation will start: Remember, that subtracting a negative number is the same as adding the positive number: A negative coordinate will have a + sign in front of it. A positive coordinate will have a − sign in front of it.

Equations That Don't Quite Look Right

Don't be confused if you see an equation which looks like this: This is still an equation of a circle, as can be seen with a little rearranging:

Note

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: This is the equation for a circle centered at the origin.

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