Cartesian → Polar formulas (x, y) → (r, θ)

A point in the plane can be written in two popular ways: Cartesian (x, y) tells you how far to move horizontally and vertically, while Polar (r, θ) tells you how far to move from the origin and at what angle. Converting between them is one of the most common “bridge skills” in algebra, trigonometry, physics, and engineering.

The radius r is the straight-line distance from (0, 0) to your point (x, y). This is literally the Pythagorean theorem:

• r = √(x² + y²)

Why? Because x and y form the legs of a right triangle, and r is the hypotenuse. Note that r is usually taken as non-negative. If r = 0, the point is the origin and θ can be any angle (because you’re not pointing anywhere).

The angle θ measures the direction of the point from the origin. You might see the basic idea written as θ = arctan(y/x), but that breaks whenever x = 0 and it can give the wrong quadrant when x and y are negative. That’s why this calculator uses:

• θ = atan2(y, x) (recommended)

The function atan2 takes both y and x and returns an angle that correctly matches the quadrant: Quadrant I gives a positive acute angle, Quadrant II gives an obtuse angle, Quadrant III returns an angle beyond 180° (or negative, depending on convention), and Quadrant IV gives a negative or near-360° angle. If your course prefers a specific range for θ (like 0–360°), use the Normalize θ option above to match that convention.

Polar → Cartesian formulas (r, θ) → (x, y)

When you know the distance and direction, you can “project” the point back onto the x- and y-axes. Imagine a right triangle where the hypotenuse is r and the angle from the positive x-axis is θ. Trigonometry tells you the adjacent side is r·cos(θ) and the opposite side is r·sin(θ).

• x = r · cos(θ)
• y = r · sin(θ)

The only “gotcha” is the unit for θ. If θ is in degrees, many calculators and software libraries still expect radians internally. This tool lets you choose the unit and handles the conversion for you. If you want the conversion yourself:

• radians = degrees × (π / 180)
• degrees = radians × (180 / π)

Also note: the same point can be written with different angles. For example, θ = 30° and θ = 390° represent the exact same direction because you’ve rotated an extra full turn (360°). The Normalize θ dropdown exists because different teachers, textbooks, and apps prefer different ranges.

Examples you can copy (and why they work)

Below are a few classic conversions that show up in homework, tests, and engineering checks. You can reproduce them using the “Try an example” button above (it rotates through examples), or type them in manually to see the steps.

This is the famous 3–4–5 right triangle. The radius is: r = √(3² + 4²) = √(9 + 16) = √25 = 5. The angle is θ = atan2(4, 3) ≈ 53.130°. So the polar form is approximately (5, 53.13°).

r = √((-1)² + 1²) = √2 ≈ 1.4142. Because x is negative and y is positive, the point is in Quadrant II. atan2(1, -1) gives an angle of 135° (or 3π/4 radians), not -45° — this is the exact reason atan2 matters. Polar form: (√2, 135°).

210° is in Quadrant III. Using x = r cos(θ), y = r sin(θ): cos(210°) = cos(180° + 30°) = -cos(30°) ≈ -0.8660, sin(210°) = -sin(30°) = -0.5. So x ≈ 10·(-0.8660) = -8.660, y ≈ 10·(-0.5) = -5.000. Cartesian form: (-8.66, -5.00) (rounded).

π/6 radians equals 30°. cos(π/6) = √3/2 and sin(π/6) = 1/2. So x = 2·(√3/2) = √3 ≈ 1.732, and y = 2·(1/2) = 1. Cartesian form: (√3, 1).

What this calculator actually does (step-by-step)

This tool follows the same workflow you’d use on paper, but it handles the tedious parts automatically and formats the answer for you.

• Reads x and y (including negatives and decimals).
• Computes r = √(x² + y²).
• Computes θ using atan2(y, x) so the quadrant is correct.
• Converts θ into degrees or radians based on your selection.
• Optionally normalizes θ into a common range (like 0–360°).
• Rounds both r and θ to your chosen decimal places.

• Reads r and θ. (r should be ≥ 0 for the standard form.)
• Converts θ to radians internally if you entered degrees.
• Computes x = r·cos(θ) and y = r·sin(θ).
• Rounds x and y to your chosen decimal places.

Finally, the calculator builds a short “Steps” log so you can see the exact substitutions and intermediate numbers (that’s helpful for homework, quizzes, and catching mistakes). If you hit Save, it stores your conversion history locally so you can compare multiple points.

Frequently Asked Questions

• Why do you use atan2 instead of arctan(y/x)?

  Because atan2(y, x) correctly handles every quadrant and the case where x = 0. The plain arctan(y/x) loses sign information and can point you to the wrong angle (especially in Quadrants II and III).

• What angle range is “right” — 0–360°, -180°–180°, 0–2π, or -π–π?

  All of them can be correct. They’re just different conventions. Many math classes prefer 0–360° (or 0–2π) for a “principal” polar angle, while some software returns -π–π. Use the Normalize θ option to match the convention you need.

• Can r be negative?

  In the standard polar form, r is non-negative. Some advanced topics allow negative r by flipping the direction (r, θ) = (-r, θ + π). For clarity, this calculator expects r ≥ 0.

• Does the unit (degrees vs radians) change the point?

  The point in space is the same — only the number you type changes. 180° equals π radians. This tool converts between them automatically so your x and y stay consistent.

• Why are my x or y values like -0.00?

  That can happen due to floating-point rounding when the true value is extremely close to zero. If you see -0.00, it’s effectively 0. Increase decimals for more detail or keep it as 0 for reporting.

• Is this only for math class?

  Not at all. Polar conversions show up in navigation, robotics, signals, graphics, and anywhere angles and distances matter. If you’ve ever used a bearing + distance, you’ve used polar thinking.

Common patterns (memorize these and you’ll fly)

If you’re studying trig, a handful of angles and triangle ratios appear constantly. Here are quick references that make conversions faster:

• 30° = π/6
• 45° = π/4
• 60° = π/3
• 90° = π/2
• 180° = π
• 360° = 2π

• cos(30°)=√3/2, sin(30°)=1/2
• cos(45°)=√2/2, sin(45°)=√2/2
• cos(60°)=1/2, sin(60°)=√3/2

Pro tip: When you convert (x, y) → (r, θ), always check the signs of x and y to confirm the quadrant matches your θ. If the signs and quadrant disagree, the angle is wrong (usually from using arctan instead of atan2).

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