Converts a point given in polar form (a distance and an angle) into its rectangular x and y coordinates.
How it works
The x-coordinate is the distance multiplied by the cosine of the angle; the y-coordinate is the distance multiplied by the sine of the angle.
What this does not include
This does not include the reverse conversion, rectangular to polar — for that, use this site’s rectangular to polar calculator instead.
How to use this calculator
- Enter the radius (r) and angle (θ) in degrees.
A worked example
r = 5, θ = 53.130102°: x = r×cos(θ) = 3, y = r×sin(θ) = 4.
r = 1.414214, θ = 45°: x = 1, y = 1.
What the variables mean
| Variable | Meaning |
|---|---|
| r | Distance from the origin |
| θ (theta) | Angle from the positive x-axis, in degrees |
Edge cases worth knowing
This is the exact reverse of this site’s cartesian-to-polar calculator — feeding its (3,4) output of r=5, θ=53.1301° back in here returns x=3, y=4, confirming the two operations undo each other.
A negative r is mathematically valid but unusual — it flips the point to the opposite direction, equivalent to adding 180° to the angle instead.
Frequently asked questions
Why does this use cosine for x and sine for y?
Because a point at angle θ and distance r from the origin traces out a right triangle where cosine gives the horizontal leg and sine gives the vertical leg.
What happens at θ = 0°?
The point lies entirely on the positive x-axis, so y becomes 0 and x equals r exactly.
Does this work for negative angles?
Yes — a negative angle is measured clockwise from the positive x-axis instead of counterclockwise, and the formula handles it correctly either way.
