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Polar-Rectangular Converter


Polar ↔ Rectangular Converter

Convert between a + bi and r∠θ
ReIm-10−10i-8−8i-6−6i-4−4i-2−2i22i44i66i88i1010iθa = 3b = 2r = 3.61z
Try these
Rectangular Form
z = a + bi
z = 3 + 2i
Polar Form
z = r · eiθ = r(cos θ + i sin 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.
The baseline triangle: legs 3 and 2, hypotenuse √13, every sign positive. Learn more about Quadrant I · All four quadrants

Getting Started — Drag, Type, or Pick a Preset
Points in All Four Quadrants
Quadrant I Baseline: 3 + 2i
Quadrant II: −4 + 3i
Quadrant III: −3 − 4i
Equal-Component States and Special Angles
Degenerate Cases — Points on the Axes
On the Imaginary Axis: 5i
On the Negative Real Axis: −6
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 aa and bb fields (rectangular) or the rr and θ\theta fields (polar) — the other form recalculates automatically. Or click one of six preset buttons below the plane: 3+2i3 + 2i, −4+3i-4 + 3i, 5i5i, −6-6, 5−5i5 - 5i, and −3−4i-3 - 4i.

Each preset places the point in a different region of the complex plane, producing a distinct triangle configuration, and each has a dedicated section below with the tool frozen on it: 3 + 2i, −4 + 3i, 5i, −6, 5 − 5i, and −3 − 4i. The Random button generates an arbitrary point within the ±10\pm 10 range. All inputs are clamped to ±10\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>0a > 0, b>0b > 0): click 3+2i3 + 2i. The green horizontal leg points right, the red vertical leg points up, and the angle θ\theta is positive between 0°0° and 90°90°. Both the real and imaginary parts are positive — this is the baseline configuration.

Quadrant II (a<0a < 0, b>0b > 0): click −4+3i-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<0a < 0, b<0b < 0): click −3−4i-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>0a > 0, b<0b < 0): click 5−5i5 - 5i. The green leg points right while the red leg drops below the real axis. The angle is negative, between 0°0° and −90°-90° — this preset is also the equal-component special case.

Each configuration is a unique visual snapshot showing how the signs of aa and bb determine the quadrant and how θ\theta changes accordingly.

Quadrant I Baseline: 3 + 2i

The preset 3+2i3 + 2i is the converter's opening state: a right triangle with nothing negative in it, and every element of the display visible at once.
ReIm-10−10i-8−8i-6−6i-4−4i-2−2i22i44i66i88i1010iθa = 3b = 2r = 3.61z
z = 3 + 2i, frozen

The bridge triangle in its home orientation: teal leg a = 3, red leg b = 2, navy hypotenuse r = 3.61, and the orange θ arc opening 33.7° from the positive real axis.

Both conversions can be read straight off the frozen frame. Rightward: r=32+22=13≈3.61r = \sqrt{3^2 + 2^2} = \sqrt{13} \approx 3.61 and θ=atan2(2,3)≈33.7°\theta = \text{atan2}(2, 3) \approx 33.7°. Leftward: 3.61cos⁡(33.7°)≈33.61 \cos(33.7°) \approx 3 and 3.61sin⁡(33.7°)≈23.61 \sin(33.7°) \approx 2 — the round trip closes, as the live conversion formulas panel confirms on every drag.

Because all signs are positive, this is the one quadrant where naive tan⁡−1(b/a)\tan^{-1}(b/a) and atan2 agree — no correction needed. Every complication this page discusses starts when the point leaves this quadrant, beginning with −4 + 3i next door.

Quadrant II: −4 + 3i

The preset −4+3i-4 + 3i crosses the imaginary axis: the real part turns negative, the triangle flips left, and the angle passes 90°90°.
ReIm-10−10i-8−8i-6−6i-4−4i-2−2i22i44i66i88i1010iθa = -4b = 3r = 5z
z = −4 + 3i, frozen

The teal leg runs left and the θ arc sweeps past the imaginary axis to 143.1°, with r = 5 — the 3-4-5 triangle mirrored into the second quadrant.

The modulus is a clean 55 — the 33–44–55 triangle mirrored into the second quadrant — and the argument is θ=atan2(3,−4)≈143.1°\theta = \text{atan2}(3, -4) \approx 143.1°, which is 180°−36.9°180° - 36.9°: the reference angle measured back from the negative real axis.

This preset is the standard trap for the naive formula: tan⁡−1(3/−4)≈−36.9°\tan^{-1}(3/-4) \approx -36.9°, a fourth-quadrant answer for a second-quadrant point. The signs of aa and bb individually — not just their ratio — decide the quadrant, which is exactly the case for atan2 made in rectangular to polar conversion.

Its double mirror is −3 − 4i: same digits, same modulus, opposite corner of the plane.

Quadrant III: −3 − 4i

The preset −3−4i-3 - 4i has both components negative — the quadrant where the sign convention for angles matters most.
ReIm-10−10i-8−8i-6−6i-4−4i-2−2i22i44i66i88i1010iθa = -3b = -4r = 5z
z = −3 − 4i, frozen

Both legs point into negative territory and the arc sweeps clockwise to −126.9°: the same 3-4-5 triangle upside down, with r still 5.

The tool reports θ=atan2(−4,−3)≈−126.9°\theta = \text{atan2}(-4, -3) \approx -126.9°, a negative angle measured clockwise from the positive real axis. The same direction could be written +233.1°+233.1°; the converter's (−180°,180°](-180°, 180°] convention always picks the representative closer to zero, so points below the real axis always read negative — compare Quadrant IV's −45°-45°.

Once again r=5r = 5: this is the third preset on the radius-5 circle, with −4 + 3i and 5i. The frozen frame shows the triangle fully inverted — legs pointing left and down, the right-angle marker tucked into the corner at (−3,0)(-3, 0) — yet built from the same two lengths as its Quadrant II mirror.

Equal-Component States and Special Angles

Click 5−5i5 - 5i to see a point where ∣a∣=∣b∣|a| = |b|. The right triangle becomes isosceles — both legs have the same length, and the angle is exactly −45°-45° (or equivalently −π4-\frac{\pi}{4} radians). The modulus is r=25+25=50≈7.07r = \sqrt{25 + 25} = \sqrt{50} \approx 7.07.

This is one of the cleanest illustrations of a 45°45° reference angle. Try typing a=5a = 5, b=5b = 5 to get the mirror image in Quadrant I at θ=45°\theta = 45°.

Other special angles are easy to produce. Setting a=5a = 5, b=53≈8.66b = 5\sqrt{3} \approx 8.66 gives θ=60°\theta = 60°. Setting a=53≈8.66a = 5\sqrt{3} \approx 8.66, b=5b = 5 gives θ=30°\theta = 30°. Each of these reference-angle triangles has a distinctive shape — tall and narrow for 60°60°, wide and flat for 30°30°, and perfectly balanced for 45°45°.

The dashed modulus circle also changes radius with each configuration, giving a visual sense of how the same modulus can map to different aa and bb combinations at different angles.
ReIm-10−10i-8−8i-6−6i-4−4i-2−2i22i44i66i88i1010iθa = 5b = -5r = 7.07z
z = 5 − 5i, frozen

An isosceles right triangle hanging below the axis: equal legs of 5, θ = −45° exactly, and the dashed modulus circle out at r = 5√2 ≈ 7.07.

The frozen isosceles frame makes the equal-legs claim checkable at a glance: both side labels read 55, the right-angle marker sits at the corner, and the orange arc stops at exactly one-eighth of a turn below the axis. In radians the polar panel reports the symbolic −π4-\frac{\pi}{4} rather than a decimal — one of the special values the polar form panel recognizes.

Equal components are the boundary between "wide" and "tall" triangles: the moment ∣b∣|b| overtakes ∣a∣|a|, the reference angle passes 45°45°. Watching the triangle cross that boundary as you drag is the fastest way to build intuition for which of the quadrant configurations you are in.

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=0a = 0 and b=5b = 5, so the green horizontal leg vanishes entirely. Only the red vertical segment remains. The modulus equals the imaginary part (r=5r = 5) and the angle is exactly 90°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=0b = 0, the red vertical leg disappears, and only the green horizontal segment remains. The modulus is 66 and the angle is 180°180°.

These axis-aligned states also demonstrate how the atan2 function handles special cases: atan2(0,−6)=180°\text{atan2}(0, -6) = 180° and atan2(5,0)=90°\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.

On the Imaginary Axis: 5i

The preset 5i5i is pure imaginary: a=0a = 0, and the whole number lives on the vertical axis.
ReIm-10−10i-8−8i-6−6i-4−4i-2−2i22i44i66i88i1010iθb = 5r = 5z
z = 5i, frozen

The triangle has collapsed to a single red vertical segment: a = 0, r = b = 5, θ = 90°. The dashed circle and quarter arc survive; the right-angle marker has nothing left to mark.

With no horizontal component, the conversion becomes trivial in one direction and instructive in the other: r=02+52=5r = \sqrt{0^2 + 5^2} = 5 and θ=atan2(5,0)=90°\theta = \text{atan2}(5, 0) = 90° exactly. Going back, a=5cos⁡(90°)=0a = 5\cos(90°) = 0 — the cosine of a right angle is what keeps the point pinned to the axis, as the polar-to-rectangular formulas spell out.

Notice that 5i5i shares its dashed modulus circle with two other presets: −4+3i-4+3i and −3−4i-3-4i all have r=5r = 5. Load the three in turn and only the orange arc moves — one circle, three angles, three different numbers. That is polar form's whole message: rr and θ\theta are independent coordinates.

The mirror case on the horizontal axis is −6, where the sine rather than the cosine does the vanishing.

On the Negative Real Axis: −6

The preset −6-6 is an ordinary negative real number — and polar form has an opinion about the minus sign: the modulus refuses to carry it.
ReIm-10−10i-8−8i-6−6i-4−4i-2−2i22i44i66i88i1010iθa = -6r = 6z
z = −6, frozen

A flat teal segment pointing left: b = 0, r = 6, and the orange arc sweeps a full half-turn to θ = 180° — the atan2 boundary case.

Polar form always keeps r≥0r \geq 0, so the negativity moves into the angle: −6=6⋅ei⋅180°-6 = 6 \cdot e^{i \cdot 180°}. The frozen frame shows the consequence — the orange arc sweeps a full half-turn from the positive axis to reach a point that rectangular form describes with a single sign.

This is also the boundary case of the argument convention. The tool reports angles in (−180°,180°](-180°, 180°], and atan2(0,−6)=180°\text{atan2}(0, -6) = 180° sits exactly on the seam — a hair above the axis would read +179.9°+179.9°, a hair below −179.9°-179.9°. The rectangular-to-polar section explains why atan2, not plain arctangent, handles this correctly.

Together with 5i, this preset completes the pair of axis degeneracies: one where the triangle loses its base, one where it loses its height.

Using the Rectangular Form Panel

The Rectangular Form section on the right shows the formula z=a+biz = a + bi and provides two input fields for aa (real part) and bb (imaginary part). Both accept values from −10-10 to 1010 in steps of 0.10.1.

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 rr and θ\theta. Below the inputs, the result line displays the complex number in standard notation — for example, 3+2i3 + 2i or −4−3i-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 ±10\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=reiθ=r(cos⁡θ+isin⁡θ)z = re^{i\theta} = r(\cos\theta + i\sin\theta) and provides input fields for rr (modulus, from 00 to about 14.1414.14) and θ\theta (argument, in degrees from −180°-180° to 180°180°).

Type a new rr 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 θ\theta and the point rotates around the origin at the current distance. Both rectangular inputs update automatically.

The result line shows rr and θ\theta in both degrees and radians. When the angle matches a well-known fraction of π\pi — such as 30°30°, 45°45°, 60°60°, 90°90°, or 180°180° — the radian display uses the symbolic form (e.g., π4\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 rr and θ\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=a2+b2r = \sqrt{a^2 + b^2} — the Pythagorean theorem applied to the right triangle. The panel substitutes the current aa and bb, computes a2+b2a^2 + b^2, and shows the final rr value.

θ=atan2(b,a)\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=rcos⁡θa = r\cos\theta — the horizontal projection. The panel substitutes the current rr and θ\theta and shows the computed aa.

b=rsin⁡θb = r\sin\theta — the vertical projection. Same substitution, showing the computed bb.

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+biz = a + bi to polar form z=reiθz = re^{i\theta} requires two calculations.

The modulus rr is the distance from the origin to the point (a,b)(a, b):

r=∣z∣=a2+b2r = |z| = \sqrt{a^2 + b^2}


The argument θ\theta is the angle measured counterclockwise from the positive real axis:

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


The atan2 function is essential because the ordinary arctangent tan⁡−1(b/a)\tan^{-1}(b/a) only returns values in (−90°,90°)(-90°, 90°) and cannot distinguish between Quadrants I and III or between Quadrants II and IV. The atan2 function uses the signs of both aa and bb to place θ\theta in the correct quadrant, returning values in (−180°,180°](-180°, 180°].

For example, z=−3−4iz = -3 - 4i gives r=9+16=5r = \sqrt{9 + 16} = 5 and θ=atan2(−4,−3)≈−126.87°\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=reiθz = re^{i\theta} back to rectangular form z=a+biz = a + bi uses basic trigonometry.

a=rcos⁡θb=rsin⁡θa = 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=5r = 5 and θ=53.13°\theta = 53.13°: a=5cos⁡(53.13°)≈3a = 5\cos(53.13°) \approx 3 and b=5sin⁡(53.13°)≈4b = 5\sin(53.13°) \approx 4, so z=3+4iz = 3 + 4i.

This conversion is a direct application of Euler's formula: reiθ=r(cos⁡θ+isin⁡θ)=rcos⁡θ+irsin⁡θ=a+bire^{i\theta} = r(\cos\theta + i\sin\theta) = r\cos\theta + ir\sin\theta = a + bi.

In the explorer, try setting r=5r = 5 and gradually changing θ\theta from 0°0° to 360°360°. The point traces a circle of radius 5, and the rectangular components oscillate as a=5cos⁡θa = 5\cos\theta and b=5sin⁡θb = 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+bia + bi) for addition and subtraction. To add z1+z2z_1 + z_2, simply add the real parts and the imaginary parts: (a1+a2)+(b1+b2)i(a_1 + a_2) + (b_1 + b_2)i. This is straightforward because addition works component-wise.

Use polar form (reiθre^{i\theta}) for multiplication, division, and powers. To multiply z1⋅z2z_1 \cdot z_2, multiply the moduli and add the arguments: r1r2⋅ei(θ1+θ2)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: (reiθ)n=rneinθ(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 and rectangular are two languages for the same point. The tools below each connect to one or both of those languages.

Euler's Formula Explorer — the theoretical backbone of polar form. Euler's identity eiθ=cos⁡θ+isin⁡θe^{i\theta} = \cos\theta + i\sin\theta is precisely why polar form is written reiθre^{i\theta} — drag θ\theta along the unit circle and watch the rectangular components cos⁡θ\cos\theta and sin⁡θ\sin\theta emerge directly.

Multiplication Visualizer — the main reason polar form exists. In rectangular form, multiplying two complex numbers requires expanding brackets; in polar form it reduces to multiplying moduli and adding angles. Use this tool after converting to polar to see why.

Division Visualizer — the inverse of multiplication: moduli divide and angles subtract. Again, polar form makes this geometric and immediate in a way rectangular form cannot.

De Moivre's Theorem Visualizer — takes polar form one step further. (reiθ)n=rneinθ(re^{i\theta})^n = r^n e^{in\theta} raises the modulus to a power and multiplies the angle — a direct consequence of the polar representation you see in this converter.

Complex Number Explorer — a general-purpose tool for plotting and manipulating complex numbers. Useful for verifying that a polar conversion lands where you expect on the plane.