Cylindrical to Cartesian coordinates calculator
A point at radius 5, angle 45° and height 3 becomes x and y of 3.535533905933 each, with z still 3. Only this direction is converted, from cylindrical to Cartesian, and the angle can be given in degrees or radians. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Radius
- 5
- Angle
- 45
- Height
- 3
- Angle unit
- Degrees
- Angle radians
- 0.7853981634
- Angle degrees
- 45
- X
- 3.5355339059
Cylindrical coordinates describe a point in three-dimensional space by how far it sits from a vertical axis, which direction it faces around that axis, and how high it is. Cartesian coordinates instead use three orthogonal displacements x, y, and z. Engineers, physics students, CNC programmers, game developers, and surveyors switch between these frames constantly: a range-and-bearing plus elevation becomes a map pin, a machine tool path becomes axis moves, and a cylindrical tank sample becomes a solid-model vertex.
What cylindrical and Cartesian coordinates mean
In the cylindrical frame a point is written (ρ, φ, z). The radius ρ is the perpendicular distance from the z-axis when ρ is non-negative; the azimuthal angle φ is measured in the xy-plane from the positive x-axis, counterclockwise for positive angles; and z is the same vertical coordinate used in Cartesian space. Cartesian coordinates write the same point as (x, y, z), where x and y are the horizontal projections and z is height.
The conversion formulas are elementary once φ is in radians: x equals ρ times cosine of φ, y equals ρ times sine of φ, and z passes through unchanged. When the angle arrives in degrees, convert first with φ_rad equals φ_deg times π divided by one hundred eighty, then evaluate cosine and sine with the host math primitives.
A classic classroom example is ρ equals five, φ equals forty-five degrees, and z equals three: both x and y equal five over square root of two (about three point five three five five) while z remains three. At φ equals zero, x equals ρ and y equals zero; at ninety degrees, x equals zero and y equals ρ. Zero radius always maps to the z-axis regardless of angle.
This tool performs only that conversion: it does not invert Cartesian to cylindrical, does not wrap angles into a principal branch, and does not plot the point in 3D.
What you can do with it
Robotics, games, and 3D animation placement
Turn range-and-bearing plus elevation into Cartesian offsets for path planning, sprites, or scene graphs without ad-hoc Math calls in every service.
CNC, survey, and engineering map pins
Batch-convert cylindrical machine or survey returns into Cartesian axis moves for CAD handoff, toolpaths, or engineering sketches.
FAQ
What formulas does the cylindrical to Cartesian calculator use?
With φ in radians: x = ρ·cos(φ), y = ρ·sin(φ), and z = z. Degrees are converted with φ_rad = φ_deg·π/180 first.
Can I enter the azimuthal angle in radians?
The default is degrees for classroom and drafting workflows.
Does this convert Cartesian coordinates back to cylindrical?
No. This capability only converts cylindrical (ρ, φ, z) to Cartesian (x, y, z). Use a Cartesian-to-cylindrical tool if you need the inverse.
Is the cylindrical to Cartesian coordinates calculator free?
The browser tool is free and runs locally.
What happens with a negative radius?
By default a negative radius is refused, so ordinary distance work stays safe. If your textbook uses signed radii, turn on the option that allows them; the point then lands on the opposite side of the axis.
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/trig/cylindrical-to-cartesian \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"radius":5,"angle":45,"height":3}'const res = await fetch("https://api.kit.forhosting.com/trig/cylindrical-to-cartesian", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"radius": 5,
"angle": 45,
"height": 3
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/trig/cylindrical-to-cartesian",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"radius": 5,
"angle": 45,
"height": 3
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/trig/cylindrical-to-cartesian", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"radius":5,"angle":45,"height":3}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"radius":5,"angle":45,"height":3}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/trig/cylindrical-to-cartesian", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"radius": 5,
"angle": 45,
"height": 3
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "trig.cylindrical_to_cartesian",
"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
This capability takes a finite cylindrical radius ρ, azimuthal angle φ—in degrees by default or in radians when you ask—and height z, and returns the Cartesian triple x equals ρ times cosine of φ, y equals ρ times sine of φ, and z unchanged as deterministic JSON. Negative radii are rejected by default so classroom distance workflows stay safe; enable allow_negative_r when you need the signed-radius convention used in some analytic geometry texts. There is no computer-algebra system, no plotter, no random seed, and no wall-clock dependency: the free browser tool and the prepaid API path share one pure solve function, so a homework check and a production script agree after fixed decimal rounding.
What cylindrical and Cartesian coordinates mean
The formulas are deterministic after at most one unit conversion—no series is re-implemented in userland, no sampling is required, and the result does not depend on locale, time zone, or network state—so graders, shaders, and API clients see the same rounded numbers every time they call the same inputs.
How to call the API and what comes back
Use it from homework pipelines, interview-prep sandboxes, robotics and game tooling, CNC and survey scripts, or any workflow that needs reproducible cylindrical-to-Cartesian conversion without installing a computer-algebra package on every machine in the lab or fleet. Send a JSON body with radius, angle, and height. Field radius is the cylindrical radial distance from the z-axis; aliases r, rho, and the ρ character map to the same meaning. Field angle is the azimuthal measure in the xy-plane; aliases phi, the φ character, theta, the θ character, and t also work. Field height is the vertical coordinate; aliases z and h are accepted. Optional angle_unit is degrees by default or radians when you pass radians (aliases unit and angle_mode). Optional allow_negative_r defaults to false; set it true only when your pipeline intentionally uses signed radii. On success the response echoes radius, angle, height, angle_unit, and allow_negative_r; adds angle_radians and angle_degrees for convenience; returns x, y, and z as rounded numbers; and includes method set to cylindrical_to_cartesian plus a short formula block documenting the definitions. Failed validation—missing radius, angle, or height, a non-finite value, an unknown unit, a magnitude past published limits, or a negative radius while negative radii are disallowed—returns invalid_input and is never charged. Pricing is a flat $0.002 per successful request with unit item.
Practical tips, negative radii, and related searches
Prefer well-known classroom triples when checking by hand: ρ equals one at zero, thirty, forty-five, sixty, and ninety degrees with a fixed height match unit-circle tables after rounding, with z unchanged. When working in radians, common angles are zero, π over six, π over four, π over three, and π over two; set angle_unit to radians so the conversion step is skipped. Keep radius non-negative unless you explicitly understand the signed-radius convention: a negative ρ with allow_negative_r true uses the same multiplications and lands at the opposite ray in the xy-plane, which is equivalent to absolute ρ at φ plus π, while z still passes through. If your data source sometimes emits negative distances by mistake, leave the default guard on and treat invalid_input as a data-quality signal rather than silently flipping the horizontal projection. Absolute radius, angle, and height magnitudes are capped so intermediate IEEE doubles stay finite on the Worker and in the browser. This landing targets people searching for a cylindrical to Cartesian coordinates calculator or a rho phi z to x y z tool in degrees or radians; it is not a Cartesian-to-cylindrical inverter and does not solve cylindrical surface equations. Failed tasks that return invalid_input do not consume prepaid balance, which keeps exploratory scripts cheap when learners probe bad radii on purpose. After a successful call, a useful sanity check is that the horizontal Euclidean length of (x, y) equals the absolute value of ρ within floating tolerance, that z equals the input height, and that the planar angle of the vector matches φ modulo two π when ρ is positive.
Integration use cases
Homework and exam checks for cylindrical–Cartesian conversion
Give students ρ, φ, and z in degrees or radians; compare their hand-computed x, y, and z against a deterministic API or free browser result.
Technical questions
What happens if the radius is negative?
By default negative radii are disallowed and the task returns invalid_input. Set allow_negative_r to true to accept signed ρ using the same formulas (equivalent to |ρ| at φ+π).