Convert a point’s polar (r, θ) coordinates into rectangular (x, y) form.
How it works
The formulas are x = r × cos(θ) and y = r × sin(θ). A point at r=5, θ=53.13° converts to (3, 4).
What this does not include
This is the reverse of this site’s Cartesian to polar calculator, which converts (x, y) coordinates into polar (r, θ) form instead.
How to use this calculator
- Enter r (the distance from the origin).
- Enter θ (the angle, in degrees).
A worked example
r = 5, θ = 53.13°: x = r×cos(θ) = 3, y = r×sin(θ) = 4.
r = 10, θ = 90°: x = 0, y = 10 — at exactly 90°, all the distance lies along the y-axis.
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
At 90°, x always comes out to zero — cos(90°) = 0, so any point at that angle sits directly on the y-axis regardless of r’s value.
A missing r makes the conversion impossible — without knowing the distance from the origin, there’s no unique (x, y) point to compute, so the calculator declines to show a result.
Why does this conversion use cosine for x and sine for y?
By convention, θ is measured from the positive x-axis, so the horizontal component of a point at distance r follows the cosine of that angle, and the vertical component follows the sine — the same relationship the unit circle illustrates directly.
Can r be negative?
In some conventions yes (pointing in the opposite direction), but this calculator uses the more common convention where r represents a non-negative distance.
What’s a common use for converting polar to Cartesian coordinates?
Any time a shape or motion is naturally described by angle and distance (like a radar reading, or circular motion) but needs to be plotted or combined using standard x-y coordinates.
