Equation of a Circle, Standard Form (Center anywhere)
A circle can be defined as the locus of all points that satisfy the equation
(xh)^{2} + (yk)^{2} = r^{2}
where r is the radius of the circle,
and h,k are the coordinates of its center.
Try this
Drag the point C and note how h and k change in the equation. Drag P and note how the radius squared changes in the equation.
Recall from
Basic Equation of a Circle,
that when the circle's center is at the origin, the formula is
When the circle center is elsewhere, we need a more general form. We add two new variables h and k that are the coordinates of the circle center point:
(xh)^{2} + (yk)^{2} = r^{2}
We subtract these from x and y in the equation to translate ("move") the center back to the origin.
If you compare the two formulae, you will see that the only difference is that the h and k variables are subtracted from the x and y terms
before squaring them:
Basic 
(x)^{2} + (y)^{2} = r^{2} 
General 
(xh)^{2} + (yk)^{2} = r^{2} 
Example
When we see the equation of a circle such as
we know it is a circle of radius 9 with its center at x = 3, y = –2.
 The radius is 9 because the formula has r^{2} on the right side. 9 squared is 81.
 The y coordinate is negative because the y term in the general equation is (yk)^{2}.
In the example, the equation has (y+2), so k must be negative: (y– (–2))^{2}
becomes (y+2)^{2}.
If the circle center is at the origin
The equation is then a little simpler. Since the center is at the origin, h and k are both zero. So the general form becomes
which simplifies down to the basic form of the circle equation:
For more on this see Basic Equation of a Circle.
Parametric form
Instead of using the Pythagorean Theorem
to solve the right triangle in the circle above, we can also solve it using trigonometry.
This produces the socalled parametric form of the circle equation as described in
Parametric Equation of a Circle.
This parametric form is especially useful in computer algorithms that draw circles and ellipses. It is described in
An Algorithm for Drawing Circles.
Things to try
 In the applet above, click 'reset' and 'hide details'.
 Drag the points C and P to create a new circle.
 Write the general formula for the resulting circle.
 Click on 'show details' to check your result.
Related topics
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