Equation of Circle in Parametric Form: Easy 2025 Guide

The equation of circle in parametric form is x = h + r cos θ and y = k + r sin θ, where (h, k) is the center, r is the radius, and θ is the parameter — an angle measured counterclockwise from the positive x-direction. As θ runs from 0 to 2π, the point (x, y) traces exactly one full circle. This form is equivalent to the standard equation (x − h)² + (y − k)² = r², but it expresses x and y separately in terms of a single angle, which makes it useful for graphing, animation, and physics problems involving rotation.

Parametric Circle Basics

A parametric equation describes a curve by giving coordinates as functions of an independent variable, called a parameter, rather than as a single relationship between x and y. For a circle, that parameter is naturally an angle.

The standard parametric equations are:

• x = h + r cos θ
• y = k + r sin θ

Here each symbol has a clear job:

• (h, k) is the center of the circle.
• r is the radius — the fixed distance from the center to every point.
• θ is the parameter, typically measured in radians, controlling which point on the circle you land on.

Why the angle works as a parameter

Picture a point on the circle. Draw a radius from the center to that point and measure the angle it makes with a horizontal line. That angle is θ. Because cos θ and sin θ always produce values between −1 and 1, the expressions h + r cos θ and k + r sin θ stay within r units of the center — exactly the definition of a circle.

Notice the naming convention: the cosine pairs with x and the sine with y. This matches the unit circle, where the point (cos θ, sin θ) sits on a circle of radius 1 centered at the origin.

Deriving the Equations

The derivation starts with the unit circle and extends it to any circle.

On the unit circle (radius 1, centered at the origin), a point at angle θ has coordinates (cos θ, sin θ). This is the fundamental definition of sine and cosine on the circle. Khan Academy covers this relationship in its trigonometry unit.

From the Pythagorean identity, cos²θ + sin²θ = 1.

Step 1 — Scale by the radius. Multiply both coordinates by r to reach a circle of radius r: the point becomes (r cos θ, r sin θ).

Step 2 — Shift the center. To move the center from (0, 0) to (h, k), add h to the x-coordinate and k to the y-coordinate, giving (h + r cos θ, k + r sin θ).

Linking back to the general equation

Now verify that this parametric point satisfies the standard circle equation (x − h)² + (y − k)² = r².

Substitute x = h + r cos θ and y = k + r sin θ:

(x − h)² + (y − k)² = (r cos θ)² + (r sin θ)²

Factor out r²: r²(cos²θ + sin²θ).

By the Pythagorean identity, cos²θ + sin²θ = 1, so the expression equals r². The point always lies on the circle — the parametric and general forms describe the same curve. Paul’s Online Math Notes provides thorough treatments of parametric equations if you want more practice with the general theory.

Worked Example

A circle is given by x = 3 + 5 cos θ, y = −2 + 5 sin θ. Find its center, radius, and standard equation, then find the point where θ = π/2.

Solution: Compare with x = h + r cos θ and y = k + r sin θ. The center is (h, k) = (3, −2) and the radius is r = 5. The standard form is (x − 3)² + (y + 2)² = 25.

For θ = π/2, cos(π/2) = 0 and sin(π/2) = 1. So x = 3 + 5(0) = 3 and y = −2 + 5(1) = 3. The point is (3, 3), the top of the circle.

Converting the other way works too. Take x² + y² − 6x + 4y − 12 = 0. Complete the square by grouping x-terms and y-terms:

(x² − 6x) + (y² + 4y) = 12

Add 9 and 4 to both sides:

(x² − 6x + 9) + (y² + 4y + 4) = 12 + 9 + 4

(x − 3)² + (y + 2)² = 25

So the center is (3, −2), r = 5, and the parametric form is x = 3 + 5 cos θ, y = −2 + 5 sin θ.

Graphing the Circle

Plotting a parametric circle is straightforward: pick values of θ, compute x and y, and connect the points. For x = 1 + 2 cos θ and y = 1 + 2 sin θ:

This traces a circle of radius 2 centered at (1, 1), moving counterclockwise. Plotting the four cardinal points alone already reveals the shape and size.

Increasing θ moves counterclockwise; replacing θ with ωt, where t is time, sets a rotation rate and makes direction and speed explicit.

Key Takeaways

• The parametric equations of a circle are x = h + r cos θ and y = k + r sin θ, with center (h, k), radius r, and parameter θ.
• They come directly from the unit circle and the Pythagorean identity cos²θ + sin²θ = 1.
• Substituting the parametric expressions into (x − h)² + (y − k)² = r² always yields r², confirming the two forms are equivalent.
• To convert from general form, complete the square to find the center and radius, then write the parametric equations.
• Plotting a handful of θ values (0, π/2, π, 3π/2) is enough to sketch the circle accurately.
• The form is especially useful for circular motion and animation paths, since one angle replaces two coordinates.