BIPM-ratified constants · v1.0

Converter

Polar, to cartesian coordinates converter calculator.

Convert polar coordinates (r, θ) to Cartesian (x, y) using x = r·cos(θ) and y = r·sin(θ). Supports degree and radian input.

From

degrees

Angle (θ)

5 degrees =2.5Cartesian Coordinate

Equivalents

Precision: 6 dp · Notation: Decimal · 3 units

Units

x-coordinatex2.5

y-coordinatey4.3301

from origin

Distancedistance5

Common pairings

1 degreesequals0.5 x

1 degreesequals0.866025 y

1 degreesequals1 distance

1 radiansequals-0.952413 x

1 radiansequals-0.304811 y

1 radiansequals1 distance

1 gradiansequals0.587785 x

1 gradiansequals0.809017 y

The conversion

How the value
is computed.

Understanding Polar to Cartesian Coordinate Conversion

Polar coordinates describe a point in a 2D plane using two values: a radial distance r from the origin and an angle θ (theta) measured counterclockwise from the positive x-axis. Cartesian coordinates describe the same point using horizontal (x) and vertical (y) distances from the origin. Converting between these two systems is fundamental in mathematics, physics, engineering, and computer graphics.

The Conversion Formulas

The polar to Cartesian converter applies two trigonometric identities derived from the unit circle definition of sine and cosine:

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

These formulas follow from right-triangle geometry. When a point sits at distance r from the origin at angle θ, its projection onto the x-axis equals r·cos(θ) and its projection onto the y-axis equals r·sin(θ). According to Paul's Online Math Notes — Calculus II: Polar Coordinates, these transformations preserve Euclidean distance from the origin, so the identity r² = x² + y² always holds after conversion.

Variables Explained

Radius (r): The non-negative distance from the origin to the point. A radius of 0 places the point at the origin regardless of θ. A radius of 5 means the point lies on a circle of radius 5 centered at the origin.
Angle (θ): The angle swept counterclockwise from the positive x-axis to the line connecting the origin to the point. Expressed in degrees (0°–360°) or radians (0–2π).
Angle Unit: Degrees or radians. When using degrees, multiply θ by π/180 before applying trigonometric functions, since sin() and cos() require radian input.
Output Coordinate: Select x (horizontal), y (vertical), or distance from origin (which equals r).

Step-by-Step Examples

Example 1: r = 5, θ = 30°

Convert polar point (5, 30°) to Cartesian:

x = 5 · cos(30°) = 5 · (√3/2) ≈ 5 · 0.8660 = 4.330
y = 5 · sin(30°) = 5 · 0.5 = 2.500

The resulting Cartesian point is approximately (4.330, 2.500).

Example 2: r = 3, θ = π/4 radians

Convert polar point (3, π/4) to Cartesian:

x = 3 · cos(π/4) = 3 · (√2/2) ≈ 3 · 0.7071 = 2.121
y = 3 · sin(π/4) = 3 · (√2/2) ≈ 3 · 0.7071 = 2.121

Since θ = 45°, the point lies on the line y = x, confirming equal x and y values of 2.121.

Angle Unit Conversion

When the input angle is in degrees, multiply by π/180 before applying trigonometric functions. For example, 60° × (π/180) ≈ 1.0472 radians. As documented in University of Nebraska's MFG Polar Coordinates reference, radian measure is most natural for calculus applications of polar coordinates, though degrees remain standard in navigation and engineering contexts.

Real-World Applications

Robotics and CNC machining: Robot arms store positions as polar coordinates (reach and rotation angle), but path-planning algorithms require Cartesian (x, y) positions for straight-line motion control.
Radar and sonar: Sensors detect objects at a range and bearing angle. Converting these readings to Cartesian grids enables integration with map-based tracking software.
Computer graphics: Circular arcs and spirals are naturally parameterized in polar form. The renderer converts each frame to Cartesian screen coordinates to draw smooth curves.
Physics: A particle on a circle of radius 2 m at 60° has Cartesian position x = 2·cos(60°) = 1.000 m and y = 2·sin(60°) ≈ 1.732 m.

Special Cases

θ = 0°: Point lies on the positive x-axis. x = r, y = 0.
θ = 90°: Point lies on the positive y-axis. x = 0, y = r.
θ = 180°: Point lies on the negative x-axis. x = −r, y = 0.
r = 0: Point is at the origin. x = 0, y = 0 for any θ.
Negative r: Reflects the point through the origin. (−3, 60°) equals (3, 240°): x = −1.500, y ≈ −2.598.

Reference

Frequently asked questions

What is the polar to Cartesian conversion formula?

The polar to Cartesian conversion uses two formulas: x = r·cos(θ) and y = r·sin(θ). Here, r is the radial distance from the origin and θ is the angle measured counterclockwise from the positive x-axis. These formulas apply to both degree and radian inputs, provided the angle is converted to radians before the trigonometric functions are applied. The identity r² = x² + y² always holds after conversion.

How do you convert polar coordinates with an angle in degrees to Cartesian form?

To convert a polar point (r, θ) where θ is in degrees, first multiply the angle by π/180 to get the radian equivalent, then apply x = r·cos(θ_rad) and y = r·sin(θ_rad). For example, the point (4, 45°) converts to x = 4·cos(45°) ≈ 2.828 and y = 4·sin(45°) ≈ 2.828, placing the point on the diagonal line y = x in the first quadrant of the Cartesian plane.

What is the difference between polar and Cartesian coordinates?

Polar coordinates locate a point using a radial distance r from the origin and an angle θ from the positive x-axis. Cartesian coordinates use two perpendicular distances — x (horizontal) and y (vertical). Polar form is natural for circular and rotational geometry, while Cartesian form is standard for algebraic equations, grid-based plotting, and linear motion. Both systems locate the same point; the best choice depends on the application at hand.

Can the radius r be negative in polar to Cartesian conversion?

By convention, a negative radius reflects the point through the origin. The polar point (−3, 60°) is equivalent to (3, 240°). Applying the formulas directly yields x = (−3)·cos(60°) = −1.500 and y = (−3)·sin(60°) ≈ −2.598, which matches the Cartesian result for (3, 240°). The conversion formula still works correctly with negative r values without any special handling required.

What are real-world applications of polar to Cartesian conversion?

Polar to Cartesian conversion is essential in robotics, where joint positions are stored as angles and reach distances but path-planning algorithms require (x, y) Cartesian data. Radar and sonar systems detect objects in polar form (range and bearing) and convert to Cartesian grids for map overlay. Computer graphics use polar parameterization for arcs and spirals, converting to screen coordinates before rendering each animation frame.

What Cartesian coordinates does the polar point (1, 0°) produce?

The polar point (1, 0°) converts to Cartesian coordinates x = 1·cos(0°) = 1·1 = 1.000 and y = 1·sin(0°) = 1·0 = 0.000. The result is the Cartesian point (1, 0), which lies directly on the positive x-axis at unit distance from the origin. This confirms that a zero-degree angle always produces a point on the positive horizontal axis, regardless of the radius value used.