Parametric Equation of a Circle

A circle can be defined as the locus of all points that satisfy the equations

x = r cos(t) y = r sin(t)

where x,y are the coordinates of any point on the circle, r is the radius of the circle and
t is the parameter - the angle subtended by the point at the circle's center.

Coordinates of a point on a circle

Looking at the figure above, point P is on the circle at a fixed distance r (the radius) from the center. The point P subtends an angle t to the positive x-axis. Click 'reset' and note this angle initially has a measure of 40°.

Using trigonometry, we can find the coordinates of P from the right triangle shown. In this triangle the radius r is the hypotenuse.

The x coordinate is therefore r cos(t) and the y coordinate is r sin(t)

To see why this is, recall that in a right triangle, the sine of an angle is the opposite side divided by the hypotenuse. In the figure on the right

In the applet above, the side opposite t has a length of y, the y coordinate of P. The hypotenuse is the radius r. Therefore Multiply both sides by r

By similar means we find that

The parametric equation of a circle

From the above we can find the coordinates of any point on the circle if we know the radius and the subtended angle. So in general we can say that a circle centered at the origin, with radius r, is the locus of all points that satisfy the equations

x = r cos(t)
y = r sin(t)

for all values of t

It also follows that any point not on the circle does not satisfy this pair of equations.

Example

If we have a circle of radius 20 with its center at the origin, the circle can be described by the pair of equations

x = 20 cos(t)
y = 20 sin(t)

What if the circle center is not at the origin?

Then we just add or subtract fixed amounts to the x and y coordinates. If we let h and k be the coordinates of the center of the circle, we simply add them to the x and y coordinates in the equations, which then become:

x = h + r cos(t)
y = k + r sin(t)

This is really just translating ("moving") the circle from the origin to its proper location. In the figure above, drag the center point C to see this.

What does 'parametric' mean?

In the above equations, the angle t (theta) is called a 'parameter'. This is a variable that appears in a system of equations that can take on any value (unless limited explicitly) but has the same value everywhere it appears. A parameter values are not plotted on an axis.

Algorithm for drawing circles

This form of defining a circle is very useful in computer algorithms that draw circles and ellipses. In fact, all the circles and ellipses in the applets on this site are drawn using this equation form. For more on this see An Algorithm for Drawing Circles.

Other forms of the equation

Using the Pythagorean Theorem to solve the triangle in the figure above we get the more common form of the equation of a circle

For more see Basic equation of a circle and General equation of a circle.

To demonstrate that these forms are equivalent, consider the figure below.

In the right triangle, we can see that

Recall the trig identity d1 Substitute x/r and y/r into the identity: Remove the parentheses: Multiply through by r²

Things to try

In the above applet click 'reset', and 'hide details'. Uncheck 'freeze radius'.
Drag P and C to make a new circle at a new center location.
Write the equations of the circle in parametric form
Click "show details" to check your answers.

Limitations

In the interest of clarity in the applet above, the coordinates are rounded off to integers and the lengths rounded to one decimal place. This can cause calculatioons to be slightly off.

For more see Teaching Notes