Rectangular to Polar Coordinates Calculator

Tip: If your θ is negative, normalization can shift it into the 0–360 range.

Rectangular coordinates (x, y) convert to polar coordinates (r, θ) using:

• r = √(x² + y²)
• θ = atan2(y, x) (handles the correct quadrant automatically)

If you select degrees, the calculator uses θ° = θ(rad) × 180 / π.

Normalization keeps the angle within a chosen interval (like 0–360° or 0–2π) while representing the same direction.

1. Enter your rectangular values x and y.
2. Choose your preferred angle unit (degrees or radians).
3. Select an angle normalization range to match your requirement.
4. Set decimal precision, then press Submit.
5. Use the download buttons to export CSV or PDF.

1. Rectangular and polar in one view

Rectangular coordinates describe a point with x and y on perpendicular axes. Polar coordinates describe the same point with a radius r from the origin and an angle θ from the positive x‑axis. This calculator converts instantly, so you can switch between graphing styles, phasor diagrams, and rotation problems without re-deriving values. In practice, r can represent magnitude in physics, impedance in circuits, or displacement in robotics. The angle indicates direction, heading, or phase, making comparisons easier across datasets. And coordinate sets.

2. Radius calculation with real inputs

The radius is computed using r = √(x² + y²). Because r is a distance, it is always non‑negative. For example, x = 3 and y = 4 gives r = 5 exactly. If x = 6 and y = −8, r = 10. Large inputs are handled with floating‑point math, and you can round the final output with a chosen precision.

3. Angle direction using atan2

The angle uses θ = atan2(y, x), which automatically selects the correct quadrant. The raw atan2 result lies in (−π, π] radians, or (−180°, 180°] in degrees. This matters when x is negative: x = −2, y = −2 produces θ = −135° even though the direction is also 225°.

4. Degrees and radians selection data

You can choose degrees or radians depending on your workflow. The conversion factor is 180/π ≈ 57.2957795, while π/180 ≈ 0.01745329252. Many CAD and surveying tasks prefer degrees, while calculus and signal processing often use radians. The unit selection keeps both r and θ consistent for reporting and exports.

5. Normalization ranges for consistent angles

Normalization helps standardize θ for charts and tables. Common intervals are 0–360° and −180–180° in degrees, plus 0–2π and −π–π in radians. If a direction returns −20°, normalizing to 0–360° changes it to 340°, which is the same ray from the origin.

6. Quadrants and axis edge cases

Quadrant labeling provides a quick sanity check. When x > 0 and y > 0 the point is in Quadrant I; x < 0, y > 0 is Quadrant II; x < 0, y < 0 is Quadrant III; and x > 0, y < 0 is Quadrant IV. If x = 0 or y = 0, the point lies on an axis.

7. Downloads for reporting and checking

Export options support documentation and sharing. CSV is useful for spreadsheets and batch checking, while the PDF summary is handy for reports or homework submissions. After calculating, download includes x, y, r, θ (raw), θ (normalized), unit, range choice, precision, quadrant, and a timestamp for traceability.

1. Why does the calculator use atan2(y, x)?

atan2 reads both x and y, so it returns the correct direction in every quadrant and handles x = 0 safely. A simple arctan(y/x) can lose quadrant information and can divide by zero.

2. What if x = 0 and y = 0?

The point is the origin, so r equals 0. The angle is mathematically undefined because every direction meets at the origin. The calculator reports the angle from atan2 and labels the location as Origin.

3. Why is my angle negative?

atan2 returns a principal angle, often in the −π to π range (or −180° to 180°). A negative value simply points clockwise from the positive x‑axis. Normalize to 0–360 or 0–2π if preferred.

4. Which normalization range should I pick?

Use 0–360° or 0–2π when you want a non‑negative heading for tables or plots. Use −180–180° or −π–π when you want angles centered around zero, common in rotation and control work.

5. Does switching degrees and radians change r?

No. The radius depends only on x and y, so r stays the same. Only θ is expressed differently, using the conversion factors 180/π and π/180. Normalization is applied within the selected unit.

6. Can I trust the CSV and PDF values?

Yes. Both exports are generated from the same stored calculation shown on the page, including your unit, normalization, precision, and timestamp. If you recalculate with new inputs, exports update to match.