Polar to Cartesian Converter
Convert polar coordinates (r, θ) to Cartesian x and y coordinates, in degrees or radians, with a step-by-step derivation and the resulting quadrant.
Input
Output
| Property | Value |
|---|---|
| No data yet | |
More ways to use this tool
REST API
curl -X POST https://api.iotools.cloud/v1/tool/polar-to-cartesian-converter \
-H "Authorization: Bearer YOUR_API_KEY" \
-H "Content-Type: application/json" \
-d '{
"radius": "5",
"angle": "45",
"angleUnit": "degrees",
"precision": "4"
}'Swap in your own key from your account. The tool's fields are the body — no wrapper.
Ask an AI agent
Use the IOTools `polar-to-cartesian-converter` tool (Polar to Cartesian Converter) on this input:
YOUR_INPUT_HEREPaste this at any agent connected to the IOTools MCP server, then add your input.
Embed widget
<iframe
src="https://iotools.cloud/embed/polar-to-cartesian-converter/"
width="100%" height="520" frameborder="0" scrolling="no" loading="lazy"
title="Polar to Cartesian Converter — iotools.cloud"
sandbox="allow-scripts allow-forms allow-same-origin allow-downloads allow-popups allow-popups-to-escape-sandbox"
allow="clipboard-write"
style="width:100%;border:1px solid #e5e7eb;border-radius:12px;overflow:hidden"></iframe>
<script src="https://iotools.cloud/embed.js" async></script>Drop this into your own page — free, no key required, just a link back.
| Cost per API/MCP call | From 5 credits |
|---|---|
| Need more credits? | View pricing |
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Guides
The Polar to Cartesian Converter turns polar coordinates (r, θ) — a distance from the origin and an angle — into Cartesian coordinates (x, y), the more familiar "over and up" pair used for graphing and most calculations. It works in either degrees or radians, and shows the exact formula substitution used to get the answer.
It's built for trigonometry and precalculus students converting homework problems, and anyone working with polar-form data (robotics headings, radar/sonar bearings, complex numbers in polar form) who needs the equivalent x/y point.
How to use it
- Enter r, the radius (distance from the origin) — this can be negative, which points in the opposite direction of θ.
- Enter θ, the angle, and choose whether it's in degrees or radians.
- Pick how many decimal places to round the result to.
- Read x and y for the Cartesian coordinates, or the Result table for those plus the radius/angle echoed back in both units, the angle normalized to 0–360°, and which quadrant (or axis) the point falls on.
- Step-by-Step Derivation shows the formula substitution.
Results update automatically as you type. For example, r = 5, θ = 45° converts to x ≈ 3.5355, y ≈ 3.5355 (Quadrant I).
What is the conversion formula?
Given a point in polar form (r, θ), the Cartesian coordinates are:
x = r · cos(θ)
y = r · sin(θ)This follows directly from the definition of sine and cosine on a circle of radius r: θ is measured counterclockwise from the positive x-axis, so r·cos(θ) is the horizontal component and r·sin(θ) is the vertical component of the point.
What does a negative radius mean?
A negative r doesn't move the calculation off the circle — it points the distance in the opposite direction from θ, which is mathematically equivalent to adding 180° to the angle. For example, (r, θ) = (-4, -30°) lands in Quadrant II, the same point you'd get from (4, 150°).
Degrees or radians?
Pick whichever your source data uses. Radians are the standard unit in most math and physics formulas (a full circle is 2π radians); degrees are more common in everyday and engineering contexts (a full circle is 360°). The calculator's θ (degrees) and θ (radians) result rows show the angle converted to both, regardless of which one you entered.
For the reverse direction — Cartesian (x, y) to polar (r, θ) — or for working with polar-form complex numbers directly, see the Complex Numbers Calculator.
Privacy
This calculator runs entirely in your browser. Your coordinates are never uploaded, logged, or stored — the computation happens locally on your device.