How do you convert rectangular form to polar form?

Compute the modulus r = √(a² + b²) using the Pythagorean theorem, then find the argument θ = atan2(b, a) to get the correct quadrant. The result is z = r·e^(iθ).

How do you convert polar form to rectangular form?

Use the projections a = r cos θ for the real part and b = r sin θ for the imaginary part. The result is z = a + bi. This follows directly from Euler's formula.

What is the difference between rectangular and polar form?

Rectangular form z = a + bi describes a complex number by horizontal and vertical components. Polar form z = r·e^(iθ) describes the same number by distance from the origin (modulus r) and angle from the real axis (argument θ). They are equivalent representations.

Why use atan2 instead of arctan for the argument?

The ordinary arctan(b/a) only returns angles between −90° and 90° and fails when a = 0. The atan2(b, a) function uses the signs of both components to return the correct angle in all four quadrants, covering the full range from −180° to 180°.

When should I use polar form vs rectangular form?

Use rectangular form for addition and subtraction since these work component-wise. Use polar form for multiplication, division, and powers since moduli multiply and arguments add, making these operations much simpler.

Polar-Rectangular Converter

Polar ↔ Rectangular Converter

Convert between a + bi and r∠θ

Try these

Rectangular Form

z = a + bi

a =b =

z = 3 + 2i

Polar Form

z = r · eiθ = r(cos θ + i sin θ)

r =θ =°

r = 3.61, θ = 33.69° = 0.19π rad

Conversion Formulas

Rectangular → Polar

r = √(a² + b²) = √(3² + 2²) = √(9 + 4) = 3.61

θ = atan2(b, a) = atan2(2, 3) = 33.69°

Polar → Rectangular

a = r cos θ = 3.61 · cos(33.69°) = 3

b = r sin θ = 3.61 · sin(33.69°) = 2

Key Ideas

Rectangular form (a + bi) describes a complex number by its horizontal and vertical components — how far right/left (a) and how far up/down (b) from the origin.

Polar form (r, θ) describes the same point by its distance from the origin (r = modulus) and the angle from the positive real axis (θ = argument).

The right triangle is the bridge. The hypotenuse is r, the adjacent side is a = r cos θ, and the opposite side is b = r sin θ. Pythagoras gives r = √(a² + b²).

When to use which? Rectangular is easier for addition and subtraction. Polar is easier for multiplication, division, and powers — because multiplying in polar means multiplying moduli and adding angles.

Getting Started — Drag, Type, or Pick a Preset

Points in All Four Quadrants

Degenerate Cases — Points on the Axes

Equal-Component States and Special Angles

Using the Rectangular Form Panel

Using the Polar Form Panel

Live Conversion Formulas

Rectangular to Polar Conversion

Polar to Rectangular Conversion

When to Use Each Form

Related Concepts and Tools

Getting Started — Drag, Type, or Pick a Preset

There are three ways to set a complex number in this converter. Drag the blue point anywhere on the complex plane and both the rectangular and polar panels update instantly. Type values directly into the

a

and

b

fields (rectangular) or the

r

and

\theta

fields (polar) — the other form recalculates automatically. Or click one of six preset buttons below the plane:

3 + 2i

-4 + 3i

5i

-6

5 - 5i

, and

-3 - 4i

.

Each preset places the point in a different region of the complex plane, producing a distinct triangle configuration. The Random button generates an arbitrary point within the

\pm 10

range. All inputs are clamped to

\pm 10

— if you type a value outside this range, a warning message appears briefly and the value snaps to the nearest limit.

Points in All Four Quadrants

Each quadrant of the complex plane produces a different triangle orientation and a different sign combination for the rectangular components.

Quadrant I (

a > 0

b > 0

): click

3 + 2i

. The green horizontal leg points right, the red vertical leg points up, and the angle

\theta

is positive between

0°

and

90°

. Both the real and imaginary parts are positive.

Quadrant II (

a < 0

b > 0

): click

-4 + 3i

. The green leg extends left of the imaginary axis while the red leg still points up. The angle is between

90°

and

180°

.

Quadrant III (

a < 0

b < 0

): click

-3 - 4i

. Both legs point in negative directions — the triangle sits below and to the left of the origin. The angle is negative, between

-90°

and

-180°

.

Quadrant IV (

a > 0

b < 0

): click

5 - 5i

. The green leg points right while the red leg drops below the real axis. The angle is negative, between

0°

and

-90°

.

Each configuration is a unique visual snapshot showing how the signs of

a

and

b

determine the quadrant and how

\theta

changes accordingly.

Degenerate Cases — Points on the Axes

When the point lies exactly on an axis, one component is zero and the triangle collapses into a line segment.

Click

5i

to place the point on the positive imaginary axis. Here

a = 0

and

b = 5

, so the green horizontal leg vanishes entirely. Only the red vertical segment remains. The modulus equals the imaginary part (

r = 5

) and the angle is exactly

90°

. There is no right-angle marker because there is no triangle — just a vertical line from the origin.

Click

-6

to place the point on the negative real axis. Now

b = 0

, the red vertical leg disappears, and only the green horizontal segment remains. The modulus is

6

and the angle is

180°

.

These axis-aligned states also demonstrate how the atan2 function handles special cases:

\text{atan2}(0, -6) = 180°

and

\text{atan2}(5, 0) = 90°

. Dragging the point along an axis lets you watch the triangle appear and disappear as the perpendicular component passes through zero.

Equal-Component States and Special Angles

Click

5 - 5i

to see a point where

|a| = |b|

. The right triangle becomes isosceles — both legs have the same length, and the angle is exactly

-45°

(or equivalently

-\frac{\pi}{4}

radians). The modulus is

r = \sqrt{25 + 25} = \sqrt{50} \approx 7.07

.

This is one of the cleanest illustrations of a

45°

reference angle. Try typing

a = 5

b = 5

to get the mirror image in Quadrant I at

\theta = 45°

.

Other special angles are easy to produce. Setting

a = 5

b = 5\sqrt{3} \approx 8.66

gives

\theta = 60°

. Setting

a = 5\sqrt{3} \approx 8.66

b = 5

gives

\theta = 30°

. Each of these reference-angle triangles has a distinctive shape — tall and narrow for

60°

, wide and flat for

30°

, and perfectly balanced for

45°

.

The dashed modulus circle also changes radius with each configuration, giving a visual sense of how the same modulus can map to different

a

and

b

combinations at different angles.

Using the Rectangular Form Panel

The Rectangular Form section on the right shows the formula

z = a + bi

and provides two input fields for

a

(real part) and

b

(imaginary part). Both accept values from

-10

10

in steps of

0.1

.

Type a value into either field and the entire visualization updates: the point moves on the plane, the triangle redraws, and the polar panel recalculates

r

and

\theta

. Below the inputs, the result line displays the complex number in standard notation — for example,

3 + 2i

-4 - 3i

.

The rectangular form is most natural for addition and subtraction. If you want to add two complex numbers, you simply add their real parts and their imaginary parts separately. This panel makes it easy to set precise integer or decimal coordinates and see how they translate into polar terms.

If you enter a value outside the

\pm 10

range, a red warning message appears: "Values are limited to ±10. Input was clamped." The message fades after a few seconds and the value is adjusted to the nearest boundary.

Using the Polar Form Panel

The Polar Form section displays

z = re^{i\theta} = r(\cos\theta + i\sin\theta)

and provides input fields for

r

(modulus, from

0

to about

14.14

) and

\theta

(argument, in degrees from

-180°

180°

).

Type a new

r

value and the point moves outward or inward along the current angle direction. The dashed circle on the plane resizes to match. Type a new

\theta

and the point rotates around the origin at the current distance. Both rectangular inputs update automatically.

The result line shows

r

and

\theta

in both degrees and radians. When the angle matches a well-known fraction of

\pi

— such as

30°

45°

60°

90°

, or

180°

— the radian display uses the symbolic form (e.g.,

\frac{\pi}{4}

) instead of a decimal approximation.

The polar form is most natural for multiplication, division, and powers. Multiplying two complex numbers in polar form means multiplying their moduli and adding their arguments. This panel lets you experiment with different

r

and

\theta

combinations and see the corresponding rectangular result.

Live Conversion Formulas

The Conversion Formulas panel shows both directions of conversion with the current values substituted in, step by step.

Rectangular → Polar displays two lines:

r = \sqrt{a^2 + b^2}

— the Pythagorean theorem applied to the right triangle. The panel substitutes the current

a

and

b

, computes

a^2 + b^2

, and shows the final

r

value.

\theta = \text{atan2}(b, a)

— the two-argument arctangent that correctly handles all four quadrants. The result appears in degrees.

Polar → Rectangular displays two lines:

a = r\cos\theta

— the horizontal projection. The panel substitutes the current

r

and

\theta

and shows the computed

a

b = r\sin\theta

— the vertical projection. Same substitution, showing the computed

b

.

As you drag the point or change any input, every number in this panel updates in real time. This makes it a live worked example — you can see the formulas at work for any configuration, not just textbook values.

Rectangular to Polar Conversion

Converting from rectangular form

z = a + bi

to polar form

z = re^{i\theta}

requires two calculations.

The modulus

r

is the distance from the origin to the point

(a, b)

r = |z| = \sqrt{a^2 + b^2}

The argument

\theta

is the angle measured counterclockwise from the positive real axis:

\theta = \text{atan2}(b, a)

The atan2 function is essential because the ordinary arctangent

\tan^{-1}(b/a)

only returns values in

(-90°, 90°)

and cannot distinguish between Quadrants I and III or between Quadrants II and IV. The atan2 function uses the signs of both

a

and

b

to place

\theta

in the correct quadrant, returning values in

(-180°, 180°]

.

For example,

z = -3 - 4i

gives

r = \sqrt{9 + 16} = 5

and

\theta = \text{atan2}(-4, -3) \approx -126.87°

. The negative angle indicates the point is below the real axis in Quadrant III.

Polar to Rectangular Conversion

Converting from polar form

z = re^{i\theta}

back to rectangular form

z = a + bi

uses basic trigonometry.

a = r\cos\theta \qquad b = r\sin\theta

The cosine projection gives the real part (horizontal component) and the sine projection gives the imaginary part (vertical component). These are exactly the legs of the right triangle shown in the visualization.

For example, given

r = 5

and

\theta = 53.13°

a = 5\cos(53.13°) \approx 3

and

b = 5\sin(53.13°) \approx 4

, so

z = 3 + 4i

.

This conversion is a direct application of Euler's formula:

re^{i\theta} = r(\cos\theta + i\sin\theta) = r\cos\theta + ir\sin\theta = a + bi

.

In the explorer, try setting

r = 5

and gradually changing

\theta

from

0°

360°

. The point traces a circle of radius 5, and the rectangular components oscillate as

a = 5\cos\theta

and

b = 5\sin\theta

— producing cosine and sine waves as functions of the angle.

When to Use Each Form

Rectangular and polar form are two representations of the same complex number. Choosing the right form simplifies the operation you need to perform.

Use rectangular form (

a + bi

) for addition and subtraction. To add

z_1 + z_2

, simply add the real parts and the imaginary parts:

(a_1 + a_2) + (b_1 + b_2)i

. This is straightforward because addition works component-wise.

Use polar form (

re^{i\theta}

) for multiplication, division, and powers. To multiply

z_1 \cdot z_2

, multiply the moduli and add the arguments:

r_1 r_2 \cdot e^{i(\theta_1 + \theta_2)}

. To divide, divide moduli and subtract arguments. To raise to a power, use De Moivre's theorem:

(re^{i\theta})^n = r^n e^{in\theta}

.

This converter lets you work in whichever form is convenient and instantly see the other. Enter an addition problem in rectangular form, read off the result, then switch to polar form for a follow-up multiplication — all without manual conversion.

Related Concepts and Tools

Polar-rectangular conversion connects to many areas of complex number mathematics. Explore these related pages for deeper coverage.

Euler's Formula Explorer — interactive visualization of

e^{i\theta} = \cos\theta + i\sin\theta

, the identity that bridges polar and exponential notation.

Complex Number Explorer — a general-purpose tool for plotting, adding, multiplying, and exploring complex numbers on the plane.

Complex Numbers — foundational theory covering the imaginary unit

i

, rectangular form, and algebraic operations.

Powers of i Calculator — compute

i^n

for any integer using the mod 4 cycle, closely related to rotation on the unit circle.

De Moivre's Theorem — extends polar-form exponentiation to arbitrary powers:

(r(\cos\theta + i\sin\theta))^n = r^n(\cos(n\theta) + i\sin(n\theta))

.

Trigonometric Identities — the cosine and sine functions used in conversion are deeply connected to angle-sum, double-angle, and other trig identities.