Every point in the plane can be described two ways: by how far right and up it is from the origin, (x, y), or by how far away it is and in which direction, (r, θ). Rectangular coordinates are natural for graphs and grids; polar coordinates are natural for anything that rotates or radiates. This converter works in both directions, keeps special-angle values exact, gets the quadrant right, and draws the point.
How to use the polar to rectangular converter
- Choose Polar → rectangular or Rectangular → polar.
- Enter r and θ, or x and y. Expressions such as
2sqrt(3)or5pi/6are accepted. - Pick the angle unit, degrees or radians.
- Read the converted point on the tape, along with the complex-number form. The diagram shows the radius, the angle arc and the projections onto the axes.
Conversion formulas
The angle is found from the reference angle tan−1|y/x| and then placed in the right quadrant:
| Quadrant | Signs of x, y | θ |
|---|---|---|
| I | +, + | reference angle |
| II | −, + | 180° − reference |
| III | −, − | 180° + reference |
| IV | +, − | 360° − reference |
Worked examples
Polar to rectangular. (r, θ) = (6, 150°). cos 150° = −√3/2 and sin 150° = 1/2, so x = 6 × (−√3/2) = −3√3 ≈ −5.196 and y = 6 × ½ = 3.
Rectangular to polar. (x, y) = (−3, 3). r = √(9 + 9) = √18 = 3√2 ≈ 4.243. The reference angle is tan−1(1) = 45°, and the point is in Quadrant II, so θ = 180° − 45° = 135° (3π/4 radians).
A Quadrant IV point. (3, −4) has r = 5 and θ ≈ 306.87°, which is the same direction as −53.13°.
Polar form of complex numbers
The same conversion links the two common ways to write a complex number. The rectangular form x + yi corresponds to the polar form r(cos θ + i sin θ), often written reiθ. In polar form, multiplying complex numbers is easy: multiply the moduli r and add the angles θ. That is why electrical engineers describe AC voltages as a magnitude and a phase angle, and why the converter lists the complex form alongside the point.
Where polar coordinates help
- Circles and spirals have simple polar equations: r = 3 is a circle, r = θ is a spiral.
- Navigation and radar report a range and a bearing — a distance and an angle.
- Rotations become additions: rotating a point by 30° just adds 30° to θ.
- Physics problems with central forces, such as orbits, separate neatly in polar form.
Keep in mind that compass bearings measure clockwise from north, while the mathematical θ measures counterclockwise from the positive x-axis. Convert with θ = 90° − bearing before using these formulas.
Related tools
For the exact sine and cosine of standard angles, use the unit circle calculator. The inverse trig functions calculator explains the range limits of tan−1, and the distance calculator computes r between any two points rather than from the origin.
Frequently asked questions
How do I convert polar to rectangular coordinates?
Use x = r cos θ and y = r sin θ. For r = 6 and θ = 150°, x = 6 × (−√3/2) = −3√3 ≈ −5.196 and y = 6 × 1/2 = 3.
How do I convert rectangular to polar coordinates?
r = √(x² + y²) and θ is the angle whose cosine is x/r and sine is y/r. For (−3, 3), r = √18 = 3√2 and θ = 135°.
Why doesn't tan⁻¹(y/x) always give the right angle?
Inverse tangent only returns angles between −90° and 90°, so it cannot tell (1, 1) from (−1, −1). For points with x < 0 you must add 180°. The calculator uses the two-argument atan2, which always picks the correct quadrant.
Are polar coordinates unique?
No. Adding 360° to θ gives the same point, and (r, θ) equals (−r, θ + 180°). The calculator reports θ in both common ranges, 0° to 360° and −180° to 180°.