Polar to Rectangular

🧭 Polar to Rectangular Converter – From r, θ to x, y

Polar coordinates describe a point in the plane using a distance from the origin (the radius r) and a direction (the angle θ). This system is natural for circular motion, antenna patterns, complex numbers, and many physics problems. Rectangular coordinates (also called Cartesian coordinates) use a horizontal value x and a vertical value y instead — the format required for most graphing, linear algebra, and engineering calculations.

📐 Conversion Formulas

The two formulas that link polar and rectangular coordinates follow directly from the definition of sine and cosine in a right triangle:

x = r · cos(θ)y = r · sin(θ)

Here r is the radial distance (always non-negative in the standard definition) and θ is measured counterclockwise from the positive x-axis. If you prefer to work with a negative radius, enable Signed Radius Handling — the tool will normalize it by flipping the angle by 180°.

🔢 Angle Unit Support

Most textbooks use degrees for introductory courses and radians for calculus and higher mathematics, but the converter also supports gradians (400 per full circle, used in surveying) and turns (1 full circle = 1 turn, convenient for signal processing). All inputs are converted to radians internally before the trigonometric evaluation.

✨ Exact Value Mode

For common angles whose trigonometric values are known surds — 0°, 30°, 45°, 60°, 90°, and all their multiples up to 360° — the tool can display the result as an exact symbolic expression such as x = r·√3/2 rather than a decimal approximation. This is useful for algebra homework, proofs, and any situation where a decimal rounded to 6 places is less useful than the exact form.

📊 Output Formats

⚡ Batch Conversion

The batch mode lets you convert many polar pairs at once — paste a list of r, θ values (one per line) and get a table of rectangular results. You can then export the table as a CSV file for use in spreadsheets or as data for graphing tools. This feature is especially helpful for physics problem sets, survey data, and programming tasks that prepare coordinate arrays.

🎨 Interactive Graph

Every single conversion produces a small Cartesian plane plot that shows:

• The radius line (green) from the origin to the converted point
• The x-projection (blue dashed) — the horizontal component
• The y-projection (red dashed) — the vertical component
• An angle arc (purple) marking the direction θ

This visual reinforces why x and y are called the rectangular projections of the polar vector.

🎓 Common Use Cases

📝 Step-by-Step Walkthrough

Example: Convert r = 10, θ = 30° to rectangular coordinates.

1. Write the formulas: x = r·cos(θ), y = r·sin(θ)
2. Substitute: x = 10·cos(30°), y = 10·sin(30°)
3. Evaluate: cos(30°) = √3/2 ≈ 0.8660, sin(30°) = 0.5
4. Result: x ≈ 8.660, y = 5.000
5. Ordered pair: (8.660, 5.000) — in Quadrant I

⚠️ Tips & Limitations

• r = 0 always gives (0, 0) regardless of angle — the origin has no defined direction.
• When the angle is a multiple of 90°, one of the outputs will be exactly 0 (or machine-epsilon close). Exact Value Mode will show 0 cleanly.
• For very large angles, the normalized angle wraps back to 0–360° (or equivalent) for readability, but the calculation uses the original angle.
• Gradians mode is common in European surveying software — a right angle is 100 grad, a full circle is 400 grad.