In mathematics, a **parametric equation** it allows to represent a curve or a surface in the plane or in the space, by means of arbitrary values or by means of a constant, called parameter, instead of using an independent variable whose values derive from those of the dependent variable. A simple example is kinematics: suppose a body that moves in a plane and, as time passes, represents a path like the one represented in Figure I. The x and y coordinates of the position of the object depend on the instant of time t . Therefore there will exist functions x and y of the variable (or parameter) t, such that x = x (t) and y = y (t); these two equations are called **parametric equations of the curve** and each value of t determines a point (x, y) on the plane.

The points (x, y) of a circle can be expressed from an individual variable θ; these individual variables are called **parameter**. The parametric equations of a circle with radius r ≥ 0 and center (h, k), are given by:

- x = h + rcosθ 0 <θ <2π
- y = k + rsinθ

The parametric equation of a circle centered at the origin and with radius r:

x^{2} + and^{2 }= r^{2}

The parametric equation with the formula of an individual variable:

- x = ± √ (r
^{2 }- Y^{2}) - y = ± √ (r
^{2 }- x^{2})

Each formula expresses a partial portion of the circle:

- y = √ (r
^{2 }- x^{2}) (cap) - y = - √ (r
^{2 }- x^{2}) (background) - x = √ (r
^{2 }- Y^{2}) (right side) - x = - √ (r
^{2 }- Y^{2}) (left)

** Examples**

- Determine the radius of the parametric equation of the circle for:

x = 5sint; y = 5sint

where 0 <t <π.

The parametric equation of a circle is x^{2} + and^{2 }= r^{2}, where x = x synt; y = y cost. Thus:

(5 synth)^{2 }+ (5 cost)^{2 }= r^{2}

25 (without^{2}t + cos^{2}t) = r^{2}

As without^{2}t + cos^{2}t = 1, we have

r^{2 }= 25

r = √25 = 5

- Determine the parametric equation of a circle with radius r = 6 and centered at the point (2,4).

We know that (h, k) = (2,4) and r = 6, therefore, the parametric equation of the circle of the statement is:

x = h + rcosθ → x = 2 + 6cosθ

y = k + rsinθ → y = 4 + 6sinθ