Polar to Rectangular Coordinates Calculator

Enter a polar coordinate (r, θ) to convert it to rectangular (Cartesian) coordinates (x, y) using x = r cos θ and y = r sin θ.

Quick Facts

x formula
x = r cos θ
The horizontal (Cartesian) coordinate.
y formula
y = r sin θ
The vertical (Cartesian) coordinate.
Reverse check
r = √(x² + y²), θ = atan2(y, x)
Converts rectangular back to polar.

Your Results

Calculated
X-coordinate
-
x = r cos θ
Y-coordinate
-
y = r sin θ
Rectangular Point
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(x, y)
Quadrant / Axis
-
Location on the Cartesian plane

Ready

Enter r and θ, choose the angle unit, then press Calculate.

Formula and Method for Polar to Rectangular Conversion

A point in the plane can be located two ways: by its Cartesian (rectangular) coordinates (x, y), or by its polar coordinates (r, θ), where r is the straight-line distance from the origin and θ is the angle measured counterclockwise from the positive x-axis. The two representations describe the same point, and you can move between them with basic right-triangle trigonometry: x = r cos θ and y = r sin θ.

How the calculation works

Picture the segment from the origin to the point as the hypotenuse of a right triangle, with length r and angle θ from the x-axis. The horizontal leg of that triangle has length r cos θ, which is the x-coordinate; the vertical leg has length r sin θ, which is the y-coordinate. This calculator converts θ to radians if you entered degrees (radians = degrees × π / 180), then computes x = r cos θ and y = r sin θ directly. It also reports which quadrant (or axis) the resulting point falls on based on the signs of x and y.

Common mistakes

  • Forgetting to convert degrees to radians: JavaScript's Math.cos and Math.sin (and most calculators' internal trig routines) expect radians — plugging degrees straight in gives a wrong answer.
  • Mixing up x and y: x always pairs with cosine and y always pairs with sine — swapping them reflects the point across the line y = x.
  • Assuming r must be positive: a negative r is valid in polar coordinates and simply points the distance r in the opposite direction of θ (equivalent to adding 180° to the angle).

Real-world applications

  • Robotics and navigation convert sensor readings (distance and bearing) into x/y positions on a map.
  • Electrical engineering converts phasor form (magnitude and phase) of AC signals into real/imaginary rectangular components.
  • Computer graphics place objects on a circular or radial layout, then convert to pixel (x, y) coordinates for rendering.
  • Physics problems involving circular motion, orbits, or rotating frames often start in polar form and need rectangular components for force or velocity analysis.

Frequently Asked Questions

What is the formula to convert polar coordinates to rectangular coordinates?
Use x = r cos θ and y = r sin θ, where r is the distance from the origin and θ is the angle measured counterclockwise from the positive x-axis. For example, r = 5 and θ = 53.13° gives x = 5 cos(53.13°) ≈ 3 and y = 5 sin(53.13°) ≈ 4, so the rectangular point is (3, 4).
Does it matter if the angle is in degrees or radians?
Yes. The sine and cosine functions expect radians internally, so if your angle is in degrees you must convert it first using radians = degrees × (π / 180). This calculator lets you pick either unit and handles the conversion for you.
Can r be negative in polar coordinates?
Yes. A negative r means the point lies in the direction opposite θ (i.e., at angle θ + 180°). The formulas x = r cos θ and y = r sin θ still apply directly and produce the correct rectangular point.
How do I know which quadrant the rectangular point falls in?
Check the signs of x and y: both positive is Quadrant I, negative x with positive y is Quadrant II, both negative is Quadrant III, and positive x with negative y is Quadrant IV. A zero value places the point on an axis rather than in a quadrant.

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