What are polar coordinates?

Polar coordinates name a point by how far it is from the origin (the radius r) and what direction it sits in (the angle θ), instead of by horizontal and vertical distance. The point (x, y) = (1, 1) in rectangular form becomes (r, θ) = (√2, 45°) in polar form: it is √2 ≈ 1.414 units away, at 45 degrees above the positive x-axis. Polar form is natural for anything that spins or radiates outward, like radar, antennas and spirals.

How do I convert (x, y) to (r, θ)?

Two formulas. The radius is r = √(x²+y²), the straight-line distance from the origin by the Pythagorean theorem. The angle is θ = atan2(y, x). For (3, 4): r = √(9+16) = √25 = 5, and θ = atan2(4, 3) ≈ 53.13°. Always use atan2 rather than atan(y/x), because atan loses the quadrant and blows up when x = 0; atan2 reads both signs and returns the right angle in all four quadrants.

Why atan2 instead of atan(y/x)?

atan(y/x) only knows the ratio, so (1, 1) and (−1, −1) both give a ratio of 1 and the same wrong answer of 45°, even though the second point is in quadrant III at −135°. It also divides by zero when x = 0. atan2(y, x) takes x and y separately, looks at both signs, and returns an angle in (−180°, 180°] that points to the real location. For (0, 1) it correctly gives 90° instead of dividing by zero.

Degrees or radians, which should I use?

Degrees split a full turn into 360 parts and read intuitively: a right angle is 90°. Radians measure the angle as arc length on a unit circle, so a full turn is 2π ≈ 6.283 and a right angle is π/2 ≈ 1.571. Calculus, programming math libraries and physics formulas expect radians; navigation, drafting and everyday work use degrees. This converter switches between them with one dropdown, so (1, 1) reads as 45° or as 0.7853982 rad.

What does converting (1, 1) give me?

(1, 1) sits in quadrant I, one unit right and one unit up. The radius is r = √(1²+1²) = √2 ≈ 1.41421, and the angle is θ = atan2(1, 1) = 45° (or π/4 ≈ 0.7854 rad). So (1, 1) in rectangular form equals (√2, 45°) in polar form. Going back, x = √2·cos45° = 1 and y = √2·sin45° = 1, which is the round-trip check the tool runs for you.

How does the angle range setting work?

atan2 returns angles in (−180°, 180°], so a point straight down comes out as −90°. The [0, 360) setting rewraps that to a single positive turn, so −90° becomes 270° and a point in quadrant III reads 225° instead of −135°. Pick [0, 360) for compass bearings and surveying where negative angles look odd, and (−180, 180] for physics and signal work where a signed phase angle is what the next formula expects.

Convert an impedance to rectangular form for series math

Your circuit gives an impedance in polar form, but adding impedances needs the resistance and reactance parts. Set polar to rectangular, enter the magnitude and angle, and read the R and X components. Now you can add it to another impedance component by component before converting the sum back to polar form.

Place a point on graph paper from a bearing

You measured a tree at 40 degrees, 12 metres from a benchmark and need its grid offset. Switch to polar to rectangular, enter r = 12 and angle = 40, and read x = 9.19 east and y = 7.71 north. The diagram shows the radius line landing in quadrant one so you can confirm the direction before marking the sheet.

Polar / Rectangular Coordinate Converter: (x,y) ⇄ (r,θ)

Convert (x, y) to (r, θ) and back, degrees or radians, with a live coordinate diagram, browser-only

Runs locally
Category Format Converter
Best for Turning pasted content or local files into a handoff-friendly format.

Convert between rectangular (Cartesian) coordinates (x, y) and polar coordinates (r, θ). Pick a direction, type the numbers, and read the answer. r = √(x²+y²) and θ = atan2(y, x); going back, x = r·cosθ and y = r·sinθ. Everything runs in your browser.

Try an example

Direction

Angle unit

Angle range

Result

r = 1.41421

θ = 45°

Quadrant: I

Every conversion is plain JavaScript that runs in this browser tab. The coordinates you type never leave the page; the shareable link encodes them in the URL only when you choose to share it.

What this tool does

Convert between rectangular (Cartesian) coordinates (x, y) and polar coordinates (r, θ) in both directions, with the angle in degrees or radians. Going to polar, the radius is r = √(x²+y²) and the angle is θ = atan2(y, x), which keeps the quadrant correct and never divides by zero when x is 0. Going back to rectangular, x = r·cosθ and y = r·sinθ. You can normalise the angle to the [0, 360) convention used in navigation and surveying, or to the (−180, 180] convention used in signal processing and physics. A small SVG diagram draws the point, the radius line and the axes so you can sanity-check the quadrant at a glance. Tiny floating-point dust like cos(90°) returning 6e-17 is snapped to zero so the answers read cleanly. One-click copy and a shareable URL reproduce your exact inputs, direction and units. Everything runs in your browser; no coordinates are uploaded.

Tool details

Input: Text + Numbers; The page exposes text boxes, numeric controls, file pickers, or structured inputs depending on the tool.
Output: Live result + Copy + Preview; The result area focuses on usable output, with copy, download, or preview actions when supported.
Privacy: Browser-side processing; The main tool logic does not call an external API, so inputs normally stay in the current tab.
Save / share: Shareable URL state; Key settings are encoded in the URL so another person can reopen the same setup.
Performance budget: Initial JS <= 9 KB; No WASM budget is declared, keeping the tool quick to open on mobile.
Best fit: Format Converter · Student; Category and role tags drive related tools, internal links, and quick fit checks.

How to use

1. Input

Paste or drop your content into the tool panel.
2. Process

Click the button. All processing is local in your browser.
3. Copy / Download

Copy the result or download to disk in one click.

How Polar / Rectangular Coordinate Converter fits into your work

Use it when the main problem is getting content from one practical format into another.

Conversion jobs

Turning pasted content or local files into a handoff-friendly format.
Previewing a conversion before you use it in a larger workflow.
Cleaning small format mismatches without opening a full editor.

Conversion checks

Try a small sample first when the source format is messy.
Check character encoding, separators, and line endings after conversion.
Keep the source until the converted output has been reviewed.

Good next steps

These links move the current task into a more complete workflow.

Real-world use cases

Convert an impedance to rectangular form for series math
Your circuit gives an impedance in polar form, but adding impedances needs the resistance and reactance parts. Set polar to rectangular, enter the magnitude and angle, and read the R and X components. Now you can add it to another impedance component by component before converting the sum back to polar form.
Place a point on graph paper from a bearing
You measured a tree at 40 degrees, 12 metres from a benchmark and need its grid offset. Switch to polar to rectangular, enter r = 12 and angle = 40, and read x = 9.19 east and y = 7.71 north. The diagram shows the radius line landing in quadrant one so you can confirm the direction before marking the sheet.

Common pitfalls

Reading r as a signed value. The radius is always zero or positive, so a negative number in the r box is not a valid polar radius. To point the other way, add 180 degrees to the angle instead.
Typing an angle into the x or y box by habit. In polar to rectangular the two inputs are magnitude and angle, not coordinates. Entering a degree value where x expects a length gives a point at the wrong distance.
Expecting an exact 0 where the math gives floating-point dust. cos(90) is mathematically 0 but raw code returns a tiny residue; this tool snaps it to 0, so do not treat a clean 0 here as a bug in your own code.

Privacy

The radius, the angle, the degree-radian switch and the SVG diagram are all computed by JavaScript inside your browser tab. The x, y, r and angle values you type stay on the page and are never sent to a server or stored. Only the share link puts your inputs into the URL, so copy the result text instead when the numbers should stay private.

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Polar / Rectangular Coordinate Converter: (x,y) ⇄ (r,θ)

What this tool does

Tool details

How to use

1. Input

2. Process

3. Copy / Download

How Polar / Rectangular Coordinate Converter fits into your work

Conversion jobs

Conversion checks

Good next steps

Real-world use cases

Convert an impedance to rectangular form for series math

Place a point on graph paper from a bearing

Common pitfalls

Privacy

FAQ

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