About the Polar to Rectangular Converter
Polar coordinates describe a point by its distance r from the origin and the angle θ it makes with the positive x-axis. Rectangular (Cartesian) coordinates describe the same point by its horizontal and vertical positions x and y. This converter goes both ways: polar to rectangular using x = r·cos θ and y = r·sin θ, or rectangular to polar using r = √(x² + y²) and the two-argument arctangent.
It is handy for precalculus and trigonometry homework, plotting polar equations, working with complex numbers in polar form, resolving vectors and forces into components, and robotics or navigation problems where a bearing and distance must become an x/y offset.
Angles are measured counter-clockwise from the positive x-axis. When converting to polar, the angle is reported both as 0°–360° and as the principal value between −180° and 180°, and the correct quadrant is always chosen — something a plain tan⁻¹(y/x) gets wrong for points on the left half of the plane.
How to use the polar to rectangular converter
- 1Choose the direction: polar to rectangular or rectangular to polar.
- 2Pick degrees or radians for the angle.
- 3Enter r and θ, or x and y.
- 4Read the converted point, plus both angle forms and the quadrant.
Formula and method
Polar to rectangular is direct trigonometry: drop a perpendicular from the point to the x-axis and you get a right triangle with hypotenuse r, so the horizontal leg is r·cos θ and the vertical leg is r·sin θ.
Rectangular to polar reverses this. The distance from the origin is the hypotenuse √(x² + y²). The angle uses atan2(y, x), the two-argument arctangent, which looks at the signs of x and y to place the angle in the correct quadrant; tan⁻¹(y/x) alone only returns angles between −90° and 90°. Results below 0° are shifted by 360° to give the 0°–360° form.
- r
- Distance from the origin (radius)
- θ
- Angle counter-clockwise from the positive x-axis
- x, y
- Rectangular (Cartesian) coordinates
Worked examples
Polar (10, 30°) to rectangular
x = 10·cos 30° = 10 × √3/2 ≈ 8.6603 and y = 10·sin 30° = 10 × ½ = 5, so the point is (8.6603, 5) in Quadrant I.
Polar (10, 2π/3 rad) to rectangular
An angle of 2π/3 radians is 120°. cos 120° = −½ and sin 120° = √3/2, so x = −5 and y ≈ 8.6603 — a point in Quadrant II.
Rectangular (−3, 4) to polar
r = √(9 + 16) = 5. Because x is negative the point is in Quadrant II, so θ = 180° − tan⁻¹(4/3) ≈ 126.87° (2.2143 rad), not the −53.13° a plain arctangent returns.
Rectangular (1, −1) to polar
r = √2 ≈ 1.4142. The point lies in Quadrant IV, 45° below the x-axis, so θ is −45° as a principal value or 315° measured all the way round.
Frequently asked questions
How do you convert polar to rectangular 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 × 0.866 ≈ 3.464.
How do you convert rectangular to polar coordinates?+
Find r = √(x² + y²), then θ = atan2(y, x). For (3, 4), r = 5 and θ ≈ 53.13°. Check the quadrant: if x is negative, add 180° to tan⁻¹(y/x).
Why does tan⁻¹(y/x) give the wrong angle sometimes?+
The ordinary arctangent only returns angles from −90° to 90°, so it cannot tell (3, 4) from (−3, −4). The atan2 function uses the signs of both x and y to return the correct quadrant.
Can a polar coordinate have a negative radius?+
Yes. A negative r means move in the opposite direction of θ, so (−r, θ) is the same point as (r, θ + 180°). Polar coordinates are not unique: adding 360° to θ also gives the same point.
Is this the same as converting a complex number to polar form?+
Yes. A complex number a + bi plots as the point (a, b), so its modulus is r = √(a² + b²) and its argument is atan2(b, a), exactly as computed here.