Basic Equation of a Circle (Center at 0,0)
A circle can be defined as the locus of all points that satisfy the equation
x^{2} + y^{2} = r^{2}
where x,y are the coordinates of each point and r is the radius of the circle.
In its simplest form, the equation of a
circle is
What this means is that for any point on the circle, the above equation will be true, and for all other points it will not.
This is simply a result of the
Pythagorean Theorem.
In the figure above, you will see a right triangle. The
hypotenuse is
the
radius
of the circle, and the other two sides are the x and y coordinates of the point P.
Applying the Pythagorean Theorem to this right triangle produces the circle equation.
As you drag the point P around the circle, you will see that the relationship between x,y and r always holds.
The radius r never changes, it is set to 20 in this applet. So x and y change according to the Pythagorean theorem
to give the coordinates of P as it moves around the circle.
Therefore, the idea here is that the circle is the
locus
of (the shape formed by) all the points that satisfy the equation.
Example
A circle with the equation
Is a circle with its center at the origin and a radius of 8. (8 squared is 64).
Solving the equation for the radius r
The equation has three variables (x, y and r). If we know any two, then we can find the third. So if we are given
a point with known x and y coordinates we can rearrange the equation to solve for r:
The negative root here has no meaning.
Note the this only works where the circle center is at the origin (0,0), because then
there is only one circle that will pass through the given point P. This finds the radius r of that circle.
Solving for a coordinate
The equation has three variables (x, y and r). If we know any two, then we can find the third. So if we are given
the radius r, and an x coordinate, we can find y by rearranging the equation:
Notice how this has two answers, due to the plus/minus.
This is expected since there are two points on the circle that have the same x coordinate.
On the right it is shown that for a given x coordinate,we see the two points p1 and p2 that share that xcoordinate.
What if the circle center is not at the origin?
Then we just add or subtract fixed amounts to the x and y coordinates to bring it back to the origin.
For more on this see
General 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 above applet click 'reset', and 'hide details'.
 Check 'Show coordinates' and uncheck 'freeze radius'.
 Drag the point P to create a circle of your choice.
 Calculate the radius of the circle, and write the equation of the circle.
 Click 'show details' to check your result.
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