Complex to Polar Converter
Convert a complex number a + bi to polar form. Enter the real part and the imaginary part to get the magnitude r and the angle theta in both degrees and radians.
Example: with Real part (a) 3 · Imaginary part (b) 4 → Magnitude r: 5.
- Angle theta (degrees)53.1301°
- Angle theta (radians)0.9273 rad
- Polar form5 ∠ 53.13° = 5(cos 53.13° + i·sin 53.13°)
Computed by the calculator below using its default values. Change any input to see your own numbers.
r = sqrt(a^2 + b^2) and theta = atan2(b, a). Using atan2 rather than plain arctangent keeps the angle in the correct quadrant across the full 360 degrees.
From a grid to a distance and a direction
A complex number a + bi is a point on a plane: a across, b up. Polar form describes that same point with a distance from the origin and a direction. The distance is the magnitude r, found with the Pythagorean theorem, and the direction is the angle theta measured from the positive real axis.
The angle needs care. Using atan2 rather than a bare arctangent tracks which quadrant you are in, so 1 + i and −1 − i do not collapse to the same 45 degrees. Once you have r and theta, multiplying and taking powers of complex numbers become simple, which is why polar form runs through so much electrical engineering and signal work.
How it’s calculated
r = sqrt(a^2 + b^2); theta = atan2(b, a). Degrees = theta × 180/pi. The polar form is r∠theta = r(cos theta + i·sin theta) = r·e^(i·theta). atan2 returns the principal argument in the range −180 to 180 degrees.
Reports the principal argument only, from −180 to 180 degrees. Add multiples of 360 degrees if you need a specific coterminal angle.
Complex numbers in polar form
| a + bi | r | theta |
|---|---|---|
| 3 + 4i | 5 | 53.13° |
| 1 + i | 1.4142 | 45° |
| 0 + 2i | 2 | 90° |
| -1 + 0i | 1 | 180° |
| 1 - i | 1.4142 | -45° |
Computed as r = sqrt(a^2 + b^2), theta = atan2(b, a); angles in degrees.
Common mistakes
- Using atan(b/a) instead of atan2, which puts second- and third-quadrant numbers in the wrong quadrant.
- Mixing degrees and radians; 45 degrees is 0.7854 rad, not 45 rad.
- Treating the magnitude as if it could be negative; r is always zero or positive, and direction is carried by the angle.
Frequently asked questions
How do I convert a complex number to polar form?
Compute the magnitude r = sqrt(a^2 + b^2) and the angle theta = atan2(b, a). The number is then r at angle theta, written r∠theta or r(cos theta + i sin theta).
What is 3 + 4i in polar form?
r = 5 and theta = 53.13 degrees, so 3 + 4i = 5∠53.13°, or about 5·e^(i·0.9273) using radians.
Why use atan2 instead of arctangent?
atan2 uses the signs of both a and b to place the angle in the correct quadrant across the full 360 degrees. Plain arctan spans only 180 degrees and mislabels many numbers.
How do I get radians from degrees?
Multiply degrees by pi/180. For example, 53.13 × pi/180 = 0.9273 radians.