Polar Form Calculator — Complex Number to Polar
Enter the real part a and imaginary part b of a complex number z = a + bi to convert it to polar form r∠θ, where r is the modulus (distance from the origin) and θ is the argument (angle from the positive real axis).
Distance from the origin in the complex plane
- 1
Sum of squares (a² + b²)
3² + 4² = 25 - 2
Modulus r = √(a² + b²)
√25 = 5The distance from the origin to (a, b) in the complex plane.
How does this calculator work?
To convert z = a + bi to polar form, compute the modulus r = √(a² + b²) and the argument θ = atan2(b, a). The polar form is r∠θ (or r(cos θ + i sin θ), or r·e^(iθ) in Euler notation). This makes multiplication of complex numbers a matter of multiplying magnitudes and adding angles.
Formula
How this is calculated
A complex number z = a + bi can be plotted on the complex plane as the point (a, b), where the horizontal axis is the real part and the vertical axis is the imaginary part. The polar form expresses z 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 r = √(a² + b²), the same formula as a 2-D Euclidean distance. The argument is found using θ = atan2(b, a), which uses both signs of a and b to place θ in the correct quadrant over the full range −180° to 180° (or −π to π radians). The polar form is then r∠θ (phasor notation) or equivalently r(cos θ + i sin θ) by the standard trigonometric identity.
The Euler form r·e^(iθ) is a compact and computationally powerful representation; it follows directly from Euler's formula e^(iθ) = cos θ + i sin θ. Polar form is especially useful when multiplying or dividing complex numbers — multiply moduli and add arguments — and in AC circuit analysis and signal processing, where phasors represent sinusoidal quantities.
Frequently asked questions
The modulus r = √(a² + b²) is the distance from the origin to the point (a, b) in the complex plane. It is always non-negative and equals the magnitude or absolute value of the complex number.
The argument θ = atan2(b, a) is the angle that the vector from the origin to (a, b) makes with the positive real axis. Using atan2 rather than plain arctan(b/a) ensures the correct quadrant for all four combinations of sign of a and b.
To multiply, multiply the moduli and add the arguments: (r₁∠θ₁) × (r₂∠θ₂) = (r₁·r₂)∠(θ₁+θ₂). This is simpler than expanding (a₁+b₁i)(a₂+b₂i) in rectangular form and is why polar form is preferred for products and powers.
Also known as
TG we-Calculate Editorial Team. (2026). Polar Form Calculator — Complex Number to Polar [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/polar-form-calculator
TG we-Calculate Editorial Team. "Polar Form Calculator — Complex Number to Polar." TG we-Calculate. 2026. https://we-calculate.com/calculator/polar-form-calculator.
TG we-Calculate Editorial Team, "Polar Form Calculator — Complex Number to Polar," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/polar-form-calculator
@misc{wecalculate_polar_form_calculator, title = {Polar Form Calculator — Complex Number to Polar}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/polar-form-calculator}}, year = {2026}, note = {TG we-Calculate} }
Did this calculator help you?