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 xaxis. 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 on the right. In the right triangle,
we can see that
Recall the trig identity
Substitute x/r and y/r into the identity:
Remove the parentheses:
Multiply through by r^{2}
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
Related topics
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