Polar to Cartesian Calculator — (r, θ) to (x, y)
Convert a point given in polar form (radius r and angle θ) to Cartesian coordinates (x, y). Works with angles in degrees or radians and handles any quadrant correctly.
Angle unit
y coordinate: 3.000007
- 1
Convert angle to radians
36.87 × π ÷ 180 = 0.643503 - 2
y = r · sin(θ)
5 × sin(0.643503 rad) = 3.000007 - 3
x = r · cos(θ)
5 × cos(0.643503 rad) = 3.999995
How does this calculator work?
To convert polar (r, θ) to Cartesian (x, y), apply x = r · cos(θ) and y = r · sin(θ). Convert degrees to radians first if needed (θ_rad = θ × π/180). The radius can be any real number (negative r reflects the point through the origin), and the angle wraps every 360°.
Formula
How this is calculated
A point in the plane can be described either in Cartesian form as (x, y) — how far right and how far up — or in polar form as (r, θ) — how far from the origin and at what angle. Converting from polar to Cartesian uses the two projection formulas x = r · cos(θ) and y = r · sin(θ), which come directly from the definition of cosine and sine in a right triangle formed by the radius and the axes.
If θ is given in degrees, the calculator first converts it to radians using θ_rad = θ × π / 180, because the trigonometric functions in computation work in radians. The radius r may be negative; a negative r points in the direction 180° opposite to θ, so the formulas still work correctly — the sign just carries through the cosine and sine.
The angle has no bounds: it wraps every 360° (2π rad), so 450° gives the same Cartesian point as 90°. The result is in the same length units as the input radius. The vector diagram shows the radial line from the origin to the resulting point, giving a visual check of the quadrant.
Frequently asked questions
The angle θ is measured counterclockwise from the positive x-axis, which is the standard mathematical convention. 0° points right, 90° points up, 180° points left, and 270° points down.
Yes. A negative radius r with angle θ gives the same point as a positive radius |r| with angle θ + 180°. The formulas x = r · cos(θ) and y = r · sin(θ) handle this correctly with the sign of r.
To go from (x, y) back to polar form, use r = √(x² + y²) and θ = atan2(y, x). The atan2 function handles all quadrants correctly, unlike plain arctan(y/x) which loses sign information.
Also known as
TG we-Calculate Editorial Team. (2026). Polar to Cartesian Calculator — (r, θ) to (x, y) [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/polar-to-cartesian-calculator
TG we-Calculate Editorial Team. "Polar to Cartesian Calculator — (r, θ) to (x, y)." TG we-Calculate. 2026. https://we-calculate.com/calculator/polar-to-cartesian-calculator.
TG we-Calculate Editorial Team, "Polar to Cartesian Calculator — (r, θ) to (x, y)," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/polar-to-cartesian-calculator
@misc{wecalculate_polar_to_cartesian_calculator, title = {Polar to Cartesian Calculator — (r, θ) to (x, y)}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/polar-to-cartesian-calculator}}, year = {2026}, note = {TG we-Calculate} }
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