Complex Number to Polar Form Calculator
Enter a complex number in rectangular form (a + bi) to get its polar form r∠θ — including the magnitude r, the angle θ in both degrees and radians, and the equivalent exponential form r·e^(iθ).
Distance from the origin in the complex plane: r = √(a² + b²)
- 1
Sum of squares
3² + 4² = 25 - 2
Argument θ
atan2(4, 3) = 53.1301°Four-quadrant inverse tangent, in degrees. - 3
Magnitude r
√25 = 5
How does this calculator work?
Given a complex number a + bi, compute its magnitude r = √(a² + b²) and argument θ = atan2(b, a). The polar form is r∠θ, and the exponential form is r·e^(iθ). These representations simplify multiplication, division, and powers of complex numbers.
Formula
How this is calculated
Every complex number a + bi corresponds to a point in the complex plane with real coordinate a and imaginary coordinate b. The polar form expresses the same point using its distance from the origin (the modulus r) and the angle it makes with the positive real axis (the argument θ).
The modulus is computed with the Pythagorean theorem: r = √(a² + b²). The argument is found using the four-quadrant inverse tangent: θ = atan2(b, a), which correctly handles all quadrants and returns a value in (−π, π] radians. Multiplying by 180/π converts to degrees.
The polar form r∠θ and the exponential form r·e^(iθ) (from Euler's formula e^(iθ) = cos θ + i·sin θ) are equivalent representations. Polar form is especially useful for multiplying or dividing complex numbers — magnitudes multiply/divide and angles add/subtract.
Frequently asked questions
Polar form r∠θ represents a complex number by its distance r from the origin (modulus) and the angle θ it makes with the positive real axis (argument). It is equivalent to the exponential form r·e^(iθ) via Euler's formula.
The standard arctan only returns angles in the range (−90°, 90°), so it cannot distinguish between a number in the first and third quadrant, or the second and fourth. The atan2(b, a) function uses the signs of both a and b to return the correct angle for all four quadrants.
The complex number 0 + 0i has zero magnitude; its argument is undefined because there is no unique direction from the origin. The calculator returns zero magnitude and 0° for this degenerate case.
TG we-Calculate Editorial Team. (2026). Complex Number to Polar Form Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/complex-number-to-polar-form-calculator
TG we-Calculate Editorial Team. "Complex Number to Polar Form Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/complex-number-to-polar-form-calculator.
TG we-Calculate Editorial Team, "Complex Number to Polar Form Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/complex-number-to-polar-form-calculator
@misc{wecalculate_complex_number_to_polar_form_calculator, title = {Complex Number to Polar Form Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/complex-number-to-polar-form-calculator}}, year = {2026}, note = {TG we-Calculate} }
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