BIPM-ratified constants · v1.0

Converter

Polar, to rectangular coordinates converter calculator.

Convert polar coordinates (r, θ) to rectangular (x, y) using x = r·cos(θ) and y = r·sin(θ). Supports degrees and radians for instant results.

From

degrees

Angle (θ)

1 degrees =0.707107Rectangular Coordinate

Equivalents

Precision: 6 dp · Notation: Decimal · 2 units

horizontal

xx0.707107

vertical

yy0.707107

Common pairings

1 degreesequals0.707107 x

1 degreesequals0.707107 y

1 radiansequals0.525322 x

1 radiansequals0.850904 y

The conversion

How the value
is computed.

What Are Polar and Rectangular Coordinates?

Polar coordinates describe a point in a plane using two values: a radial distance r from the origin and an angle θ measured counterclockwise from the positive x-axis. Rectangular (Cartesian) coordinates describe the same point using a horizontal distance x and a vertical distance y from the origin. A polar to rectangular converter makes switching between these systems fast and error-free.

The Conversion Formula

Converting from polar to rectangular coordinates relies on trigonometric identities derived from the unit circle. For any point at radius r and angle θ, the rectangular coordinates are:

x = r · cos(θ) — the horizontal component
y = r · sin(θ) — the vertical component

Formula Derivation

The derivation follows directly from right triangle trigonometry. A point P at distance r from the origin forms a right triangle with the x-axis. By definition, the cosine of θ equals the adjacent side divided by the hypotenuse, and the sine of θ equals the opposite side divided by the hypotenuse. Setting the hypotenuse equal to r yields x = r·cos(θ) and y = r·sin(θ). This derivation is detailed in Paul's Online Math Notes — Calculus II: Polar Coordinates and confirmed by the University of Maryland Calculus 141 Polar Coordinates lecture notes.

Understanding the Variables

r (Radius): The radial distance from the origin to the point. Any non-negative real number is valid. A radius of 0 always maps to the origin (0, 0) regardless of angle.
θ (Theta): The angle measured counterclockwise from the positive x-axis. Expressed in degrees (0° to 360°) or radians (0 to 2π ≈ 6.2832). Common reference angles: 0° maps to (1, 0), 90° to (0, 1), 180° to (-1, 0), and 270° to (0, -1).
Angle Unit: Specifies whether θ is entered in degrees or radians. Multiply degrees by π/180 to convert to radians; multiply radians by 180/π to convert to degrees.
Output Coordinate: Selects which rectangular value to return — the x component or the y component.

Worked Examples

Example 1: r = 5, θ = 30°

x = 5 · cos(30°) = 5 · 0.8660 ≈ 4.33

y = 5 · sin(30°) = 5 · 0.5000 = 2.50

Rectangular coordinates: (4.33, 2.50).

Example 2: r = 10, θ = π/4 radians (45°)

x = 10 · cos(π/4) = 10 · (√2/2) ≈ 7.07

y = 10 · sin(π/4) = 10 · (√2/2) ≈ 7.07

Rectangular coordinates: (7.07, 7.07). Both components are equal because 45° bisects the first quadrant symmetrically.

Example 3: r = 3, θ = 270°

x = 3 · cos(270°) = 3 · 0 = 0

y = 3 · sin(270°) = 3 · (-1) = -3

Rectangular coordinates: (0, -3), a point directly below the origin on the negative y-axis.

Understanding Quadrants in Polar-to-Rectangular Conversion

The rectangular coordinate system divides the plane into four quadrants defined by the signs of the x and y coordinates. Understanding which quadrant a converted point occupies is essential for verifying your calculations and visualizing the point's location. The first quadrant spans 0° to 90°, where both x and y are positive. The second quadrant spans 90° to 180°, where x is negative and y is positive. The third quadrant spans 180° to 270°, where both x and y are negative. The fourth quadrant spans 270° to 360°, where x is positive and y is negative. This quadrant system provides a powerful sanity check: if your polar angle should place a point in the second quadrant, but your converted rectangular coordinates show positive x, an error has occurred. For example, converting the polar coordinates (5, 120°) should yield a point in the second quadrant. Computing x = 5·cos(120°) = 5·(-0.5) = -2.5 and y = 5·sin(120°) = 5·(0.866) ≈ 4.33 confirms this expectation, with negative x and positive y values placing the point correctly in the second quadrant.

Real-World Applications

Navigation and GPS: Converting polar bearings and distances to x-y grid coordinates in mapping systems.
Robotics: Translating LIDAR or sonar range-and-angle readings into Cartesian workspace coordinates for path planning.
AC Circuit Analysis: Converting impedance magnitude and phase angle into resistance (real) and reactance (imaginary) components.
Physics: Decomposing force and velocity vectors expressed as magnitude and direction into horizontal and vertical components.
Computer Graphics: Placing objects whose positions are defined by radius and angle into screen pixel coordinate systems.

Reference

Frequently asked questions

What is the formula to convert polar coordinates to rectangular coordinates?

The conversion formulas are x = r·cos(θ) and y = r·sin(θ), where r is the radial distance from the origin and θ is the angle measured from the positive x-axis. For example, the polar point (r = 5, θ = 60°) converts to x = 5·cos(60°) = 2.50 and y = 5·sin(60°) ≈ 4.33, giving the rectangular coordinates (2.50, 4.33). These identities follow directly from right triangle trigonometry and the unit circle definition.

How do I convert polar coordinates when the angle is in degrees versus radians?

When θ is in degrees, plug the degree value directly into cos(θ°) and sin(θ°). When θ is in radians, use the radian value instead — for example, θ = π/3 radians equals 60°, so cos(π/3) = 0.5. To switch units, multiply degrees by π/180 to get radians, or multiply radians by 180/π to get degrees. This converter accepts both units and handles the distinction automatically, eliminating manual unit conversion errors.

What does a negative radius mean when converting polar to rectangular coordinates?

A negative radius reflects the point through the origin. The polar pair (-r, θ) is equivalent to (r, θ + 180°). For example, (-4, 45°) converts to x = -4·cos(45°) ≈ -2.83 and y = -4·sin(45°) ≈ -2.83, placing the result in the third quadrant — directly opposite the location a positive r = 4 at 45° would produce. Some textbooks restrict r to non-negative values, but many contexts allow negative radii.

Can every polar coordinate pair be converted to rectangular form?

Yes, every polar pair (r, θ) has a valid rectangular equivalent. The conversion is always defined because cosine and sine return real numbers for any angle input. A special case is r = 0, which always produces the origin (0, 0) regardless of θ. Angles outside the standard 0°–360° range, such as θ = 450°, are treated as their coterminal equivalents (450° = 90°), so x = r·cos(90°) = 0 and y = r·sin(90°) = r.

What is the difference between polar and rectangular (Cartesian) coordinates?

Rectangular coordinates locate a point by measuring horizontal (x) and vertical (y) distances along two perpendicular axes, making them ideal for linear equations and standard graphing. Polar coordinates locate the same point using a radial distance r and an angle θ, which is more natural for circular motion, spiral curves, and wave functions. Converting between the two systems is essential whenever data collected in one form must be analyzed or visualized in the other.

What are the most common real-world applications of polar to rectangular conversion?

Polar to rectangular conversion is used in radar and sonar systems (converting range and bearing to map x-y positions), robotics (transforming sensor distance-angle readings to workspace coordinates), AC electrical engineering (converting impedance magnitude and phase to resistance and reactance), GPS and surveying, computer graphics (mapping angular positions to screen pixels), and physics (decomposing force and velocity vectors given as magnitude-angle pairs into horizontal and vertical components for equations of motion).