Equation of a Circle, Standard Form (Center anywhere)
A circle can be defined as the locus of all points that satisfy the equation
(x-h)2 + (y-k)2 = r2
where r is the radius of the circle,
and h,k are the coordinates of its center.
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.
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:
(x-h)2 + (y-k)2 = r2
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:
||(x)2 + (y)2 = r2
||(x-h)2 + (y-k)2 = r2
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 r2 on the right side. 9 squared is 81.
- The y coordinate is negative because the y term in the general equation is (y-k)2.
In the example, the equation has (y+2), so k must be negative: (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.
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 so-called 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.
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