The LessonThe 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.
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:
Lesson SlidesThe slider below explains why the "Equation of a Circle" works. Open the slider in a new tab
The Center of a Circle Has Negative CoordinatesThe 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.
(x − −1)2 + ...Remember, that subtracting a negative number is the same as adding the positive number:
(x − −1)2 = (x + 1)2A 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 RightDon't be confused if you see an equation which looks like this:
(x − 1)2 + (y − 3)2 − 49 = 0This is still an equation of a circle, as can be seen with a little rearranging:
(x − 1)2 + (y − 3)2 = 49
Circle Centered at the OriginA 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 = r2This is the equation for a circle centered at the origin.