Polar form of a complex number
3 + 4i has a magnitude of 5 and an angle of about 0.927 radians, so its polar form is 5∠0.927. Enter the real and imaginary parts to get the magnitude, the angle in the right quadrant and the cis and phasor forms. Free in your browser.
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Worked example
- Real
- 3
- Imag
- 4
- Magnitude
- 5
- Angle
- 0.927295218
- Rectangular
- 3+4i
- Polar
- 5∠0.927295218002
A complex number written in rectangular form as a plus bi is a pair of real coefficients, but the same number can be rewritten as a length and a directed angle. That rewrite is the polar form of a complex number: the magnitude is the modulus, the Euclidean distance from the origin in the Argand plane, and the angle is the principal argument in radians, measured from the positive real axis. Multiplication, division, integer powers, and roots collapse to products and sums of those two scalars, which is why every algebra course and every phasor notebook eventually asks for the polar form of a plus bi. This polar form calculator accepts the two finite real and imaginary parts of one rectangular operand, computes the magnitude with a numerically stable hypotenuse, and computes the argument with a two-argument arctangent so the angle lands in the correct quadrant.
What polar form records: modulus, argument, and the cis identity
A complex number in rectangular form is the pair a plus bi, where a is the real part, b is the imaginary part, and i is the imaginary unit with i squared equal to negative one. Plotting that pair as a point in the Argand plane makes the polar rewrite obvious: the magnitude r is the length of the segment from the origin to the point, and the angle theta is the directed turn from the positive real axis to that segment.
Once r and theta are known, Euler's identity writes the same number as r times e to the i theta, and the trigonometric form writes it as r times the quantity cosine theta plus i sine theta, often abbreviated r cis theta. Engineering worksheets also print the compact phasor r angle theta. All of those strings describe one number; they differ only in notation.
Zero real or zero imaginary parts are allowed: a pure real number has argument zero or plus or minus pi, and a pure imaginary number has argument plus or minus pi over two. The origin zero plus zero i is a finite pair, so this calculator returns magnitude zero and angle zero rather than inventing an undefined-argument failure that the contract does not promise.
The formula field always restates the identity magnitude equals hypot of real and imag, argument equals atan2 of imag and real, so a tutorial can show the rule next to the numeric answer without keeping a second content table.
The two parts you enter and the forms you get
On success the body echoes the two rounded input coefficients, returns magnitude and angle as numbers rounded to twelve decimal places for stable goldens across runtimes, and supplies three display strings: rectangular for the original a plus bi form, polar for the compact r angle theta phasor, and cis for the trigonometric r times cosine plus i sine expansion. Example: real three and imag four yields magnitude five and angle atan2 of four and three, about zero point nine two seven radians, with polar string five angle that same argument.
Angle is always in radians, never degrees, because the polar form of a complex number in analysis and in most programming libraries is defined on the real line modulo two pi, not on a three-hundred-sixty-degree circle. Pair the response with your own Argand-plane plot if you need a diagram; this capability converts one rectangular operand to polar form only and does not multiply, divide, or raise the result to a power.
Where polar form saves work
Classrooms and tutoring platforms use a polar form of a complex number calculator so learners can check the modulus and the principal argument without flipping the quadrant when a is negative. Signal-processing and control scripts often store a plus bi samples and need the modulus for energy and the argument for phase unwrapping later in the pipeline. Failed validation does not consume balance.
Keep requests small, pin magnitude and angle in tests, and you get reproducible polar form for homework, grading, and edge workers without standing up symbolic math infrastructure of your own.
What you can do with it
Classroom and homework checks
Let students submit the real and imaginary parts of a plus bi and verify the modulus and principal argument against textbook polar form.
Exam and LMS auto-grading
Mark polar-form items with a deterministic oracle that rejects non-finite parts and always returns the same radian argument for a given pair.
Phasor and impedance notebooks
Convert rectangular impedances or phasors to magnitude and angle before multiplying transfer functions, without importing a computer-algebra library.
FAQ
How do I convert a complex number from rectangular form to polar form?
Send two finite numbers: real and imag for z = real + imag·i.
Is the angle returned in degrees or radians?
Radians only. The principal argument uses atan2 and lies in (−π, π]. Convert to degrees yourself by multiplying by 180/π if a worksheet asks for degrees.
What polar form is returned for a negative real number, such as −1 + 0i?
Magnitude 1 and angle π (about 3.14159265359 radians). The two-argument arctangent places the result on the negative real axis instead of reporting zero from a naïve one-argument arctangent of imag/real.
What happens if real or imag is missing, infinite, or not a number?
Invalid input is not billed and does not return a partial polar form.
Why write a complex number in polar form at all?
Because multiplication and division become simple: magnitudes multiply and angles add. Powers and roots follow the same pattern, which is why phasor calculations in electrical engineering switch to polar form before combining values.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/math/polar-form \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"real":3,"imag":4}'const res = await fetch("https://api.kit.forhosting.com/math/polar-form", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"real": 3,
"imag": 4
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/math/polar-form",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"real": 3,
"imag": 4
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/math/polar-form", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"real":3,"imag":4}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"real":3,"imag":4}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/math/polar-form", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"real": 3,
"imag": 4
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "math.polar_form",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
Missing or non-finite parts fail with invalid_input and are not billed.
What polar form records: modulus, argument, and the cis identity
Algebraically r equals the square root of a squared plus b squared, which this endpoint evaluates with a hypotenuse so overflow-prone intermediate squares stay well behaved, and theta equals the two-argument arctangent of b and a so the result sits in the open-closed interval from negative pi to plus pi rather than in a single first-quadrant arctangent that would scramble signs.
Input contract, response shape, and validation
POST a JSON body with two required finite numbers: real, the coefficient a of the rectangular operand, and imag, the coefficient b. Each field may arrive as a JSON number or as a numeric string; empty strings, missing keys, Infinity, NaN, and non-numeric text fail with invalid_input naming the field, and those failures do not consume prepaid balance. Optional aliases such as a, re, and real_part map onto real, and b, im, imag_part, and imaginary map onto imag, so generated clients and hand-written scripts stay flexible.
Where a polar-form API earns its keep
Electrical-engineering notebooks convert rectangular impedances and phasors into polar form before multiplying transfer functions, because magnitudes multiply and angles add; calling a tiny API avoids pulling a full computer-algebra system for two floating-point operations. Because the implementation is pure parse, every successful call costs the published $0.002 base price.
Technical questions
What does the polar form calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.