Polar Form Calculator

Convert a complex number a + bi (or a point (a, b)) to polar form r(cos θ + i sin θ) — enter the real and imaginary parts to get the magnitude and angle.

Quick Facts

Magnitude (modulus)
r = √(a² + b²)
The distance from the origin to the point (a, b).
Angle (argument)
θ = atan2(b, a)
Four-quadrant arctangent; correctly signed for all a, b.
Polar form
z = r(cos θ + i sin θ) = r·e^(iθ)
Also written in shorthand as z = r cis θ or z = r∠θ.

Your Results

Calculated
Magnitude (r)
-
r = √(a² + b²)
Angle (θ) in degrees
-
θ = atan2(b, a) × 180/π
Angle (θ) in radians
-
θ = atan2(b, a)
Polar form
-
z = r(cos θ + i sin θ)

Ready

Enter the real and imaginary parts, then press Calculate.

Formula and Method for Converting to Polar Form

Every complex number a + bi (or, equivalently, every point (a, b) in the plane) can be written in rectangular form using its horizontal and vertical components, or in polar form using its distance from the origin and its direction. The polar form is z = r(cos θ + i sin θ), where r is the magnitude (also called the modulus) and θ is the angle (also called the argument), measured counterclockwise from the positive real axis. This calculator takes a and b and returns r and θ, plus the assembled polar expression.

How the calculation works

The magnitude is found with the Pythagorean theorem, since a, b, and r form a right triangle: r = √(a² + b²). The angle is found with the two-argument arctangent function, written atan2(b, a), which returns the correct angle in every quadrant: θ = atan2(b, a). Ordinary arctan(b/a) is not enough on its own — it cannot distinguish, for example, (3, 4) from (−3, −4), and it is undefined when a = 0 — while atan2 uses the sign of both a and b to place θ in the correct quadrant, in the range (−180°, 180°] or (−π, π]. Once r and θ are known, the polar form is assembled as z = r(cos θ + i sin θ), which is equivalent to the exponential (Euler) form z = r·e^(iθ) and the shorthand notation r∠θ or r·cis θ.

Common mistakes

  • Using arctan(b/a) directly: this gives the wrong quadrant whenever a is negative — always use the four-quadrant atan2(b, a) instead.
  • Mixing degrees and radians: cos θ and sin θ in most programming languages expect radians; convert with θ(rad) = θ(deg) × π/180 before using trig functions if your angle is in degrees.
  • Forgetting the origin case: the complex number 0 + 0i has magnitude 0 but no defined angle, since every direction is equally valid at the origin.

Real-world applications

  • Electrical engineering uses polar form to represent AC voltage and current phasors, making multiplication and division of impedances far simpler than in rectangular form.
  • Physics uses polar (and the related cylindrical/spherical) coordinates for problems with rotational symmetry, such as orbital motion or wave propagation.
  • Signal processing and control theory use the polar form of complex numbers (magnitude and phase) to analyze frequency response.
  • Navigation and robotics use magnitude-and-angle (r, θ) pairs directly, since heading and distance are more natural than x/y offsets.

Frequently Asked Questions

What is the polar form of a complex number?
The polar form expresses a complex number a + bi as z = r(cos θ + i sin θ), where r = √(a² + b²) is the magnitude (modulus) and θ = atan2(b, a) is the angle (argument) measured counterclockwise from the positive real axis. It can also be written using Euler's formula as z = r·e^(iθ).
How do you find r and θ from a and b?
Compute the magnitude with the Pythagorean theorem: r = √(a² + b²). Compute the angle with the two-argument arctangent: θ = atan2(b, a), then convert to degrees by multiplying by 180/π if needed.
Why use atan2(b, a) instead of arctan(b/a)?
arctan(b/a) alone cannot tell which quadrant the point (a, b) is in and is undefined when a = 0. atan2(b, a) uses the signs of both a and b to return the correct angle in the full range (−180°, 180°], including on the axes.
How do you convert a polar form number back to rectangular form?
Use a = r·cos θ and b = r·sin θ to recover the real and imaginary parts from the magnitude r and angle θ.

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