Enter a radius r and an angle θ in degrees or radians (π is fine, like 3π/4) to get the rectangular coordinates (x, y), with exact values for common angles, the working and the point plotted on a polar grid. Paste a list to convert many points at once, or check the reverse conversion.
Last updated:
Mode
Accepts numbers and π, e.g. 2π/3, pi/4, -1.2
Angle unit
Separate r and θ with a comma, space or tab. θ uses the angle unit above.
Enter your values to see the result.
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How to use the Polar to Cartesian Converter
Enter the radius r (the distance from the origin; it may be negative).
Enter the angle θ and choose degrees or radians. You can type π or pi, e.g. 5π/6.
Read x and y, plus the exact form for common angles.
Check the plot and steps to see the point and how it was found.
For many points, switch to Bulk list, paste one "r, θ" pair per line and download the CSV.
Polar to rectangular formulas
Conversion
Formula
Example
Polar → x
x = r·cos θ
r = 4, θ = 60° → x = 2
Polar → y
y = r·sin θ
r = 4, θ = 60° → y = 2√3 ≈ 3.4641
Cartesian → r
r = √(x² + y²)
(2, 3.4641) → r = 4
Cartesian → θ
θ = atan2(y, x)
(2, 3.4641) → θ = 60°
Degrees → radians
× π/180
60° = π/3 ≈ 1.0472
Exact values for common angles (r = 1)
Multiply both coordinates by r for any other radius. Signs follow the quadrant: x is negative in quadrants II and III, y is negative in quadrants III and IV.
θ (degrees)
θ (radians)
x = cos θ
y = sin θ
0°
0
1
0
30°
π/6
√3/2 ≈ 0.8660
1/2
45°
π/4
√2/2 ≈ 0.7071
√2/2 ≈ 0.7071
60°
π/3
1/2
√3/2 ≈ 0.8660
90°
π/2
0
1
120°
2π/3
−1/2
√3/2
135°
3π/4
−√2/2
√2/2
180°
π
−1
0
270°
3π/2
0
−1
Common mistakes
Wrong angle mode. cos 60 in radian mode is −0.952, not 0.5. Check whether θ is in degrees or radians before converting.
Using arctan for the reverse. arctan(y ÷ x) cannot tell (1, 1) from (−1, −1). Use atan2 or add 180° when x is negative.
Measuring θ from the wrong axis. In math, θ starts at the positive x-axis and turns counterclockwise. Compass bearings start at north and turn clockwise, so convert them first: θ = 90° − bearing.
Frequently asked questions
How do you convert polar coordinates to Cartesian coordinates?
Use x = r·cos θ and y = r·sin θ. For r = 4 and θ = 60°: x = 4 × 0.5 = 2 and y = 4 × 0.8660 ≈ 3.4641, so the point is (2, 2√3).
Do I need degrees or radians?
Either works, as long as your calculator uses the same mode as the angle. 180° = π radians, so multiply degrees by π/180 to get radians. This converter lets you choose and accepts π in radian input.
What does a negative r mean?
A negative radius points in the opposite direction of the angle. The point (−3, 45°) is the same as (3, 225°), which is (−2.1213, −2.1213).
How do I convert Cartesian to polar coordinates?
r = √(x² + y²) and θ = atan2(y, x). Use atan2 (or adjust arctan(y ÷ x) by adding 180° for points in quadrants II and III) to get the right quadrant. The Cartesian → polar mode in this tool does this.
Is a polar point unique?
No. Adding any multiple of 360° (2π) to θ gives the same point, and (r, θ) is also the same as (−r, θ + 180°). Cartesian coordinates, by contrast, are unique for each point.
Why does my answer show a tiny number like 0.0000 instead of 0?
Values like cos 90° are exactly 0, but computers store π approximately, so the result can be something like 6 × 10⁻¹⁷. The converter rounds to the decimal places you pick, so these show as 0.
Where are polar coordinates used?
In navigation and radar (bearing and distance), robotics and physics (circular motion), complex numbers (modulus and argument), and in graphing curves like circles, spirals and roses that are simpler in polar form.
Are all the tools on FreeToolCabin tested for quality and accuracy?
Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.