Polar to Complex Calculator

This polar to complex calculator converts a complex number from polar form, given by a modulus r and an angle theta, into rectangular form a plus bi. Enter the modulus and the angle in degrees and the calculator returns the real and imaginary parts and the full rectangular number. Polar form is ideal for multiplying, dividing and taking powers of complex numbers, while rectangular form is needed for adding and subtracting them, so converting between the two is a routine task. The conversion uses the cosine and sine of the angle. The formula, a worked example and the assumptions are below.

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Modulus r

Angle theta (degrees)

Imaginary part b

Rectangular form

The formula

From polar form r and theta, the rectangular parts are a = r cos(theta) and b = r sin(theta), giving the complex number a + bi. This is the reverse of taking the modulus and argument.

Worked example

For modulus 5 and angle 53.13 degrees, a = 5 cos(53.13) = 3 and b = 5 sin(53.13) = 4, so the number is 3 + 4i. Enter 5 and 53.130102 to confirm.

Frequently asked questions

How do I convert polar to rectangular?

Multiply the modulus by the cosine of the angle for the real part, and by the sine for the imaginary part.

Why convert to rectangular?

Rectangular form a + bi is needed to add or subtract complex numbers; polar form is better for multiplying and powers.

What units is the angle in?

This calculator takes the angle in degrees and converts internally to radians.

Who this calculator is for

This calculator is for algebra students and anyone needing a quick, reliable result.

What this calculator assumes

You enter valid numbers.
The standard method is applied.
Results are rounded for display.

Formula and sources

Related calculators

Quadratic Formula Calculator: solve a quadratic.
Scientific Calculator: general maths.
Matrix Calculator: matrix operations.

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