Polar to Cartesian Calculator
Convert polar coordinates to x, y coordinates.
Results update as you type.
How it works
Polar and Cartesian coordinates are two ways to describe the same point, and this calculator converts from polar to Cartesian. You enter the radius, the point's distance from the origin, and the angle in degrees measured from the positive x-axis, and it returns the x and y coordinates. The conversion uses basic trigonometry: the x-coordinate is the radius times the cosine of the angle, and the y-coordinate is the radius times the sine of the angle. This works because the radius and angle define the hypotenuse and direction of a right triangle whose horizontal and vertical legs are exactly the x and y values. Polar coordinates are natural for describing circular or rotational situations, such as points on a radar screen, angles and distances in navigation, or the position of a robot arm, while Cartesian coordinates are needed for plotting on a standard grid, most computer graphics, and many calculations. Converting between them is a routine but essential task in mathematics, physics, engineering, and computer graphics. The angle here is in degrees for convenience; the calculator converts it to radians internally as the trigonometric functions require. A radius can be positive or, by convention, negative, which points in the opposite direction. Enter the radius and angle to convert polar coordinates to Cartesian x and y.
x = r·cos(θ) ; y = r·sin(θ)
Polar coordinates give a point by its distance r from the origin and its angle θ from the positive x-axis. To convert to Cartesian coordinates, the x-value is r times the cosine of the angle and the y-value is r times the sine of the angle, with the angle here taken in degrees and converted to radians internally.
Source:Coordinate geometry
We review all our tools from time to time to ensure accuracy.
Worked example
How to use this calculator
- 1
Enter your values
Enter radius (r) and angle (θ).
- 2
Adjust the options
Fine-tune any options to match your situation.
- 3
Read your result
x-coordinate updates instantly as you type — no button to press.
Frequently asked questions
How do you convert polar to Cartesian coordinates?
Multiply the radius by the cosine of the angle to get the x-coordinate, and multiply the radius by the sine of the angle to get the y-coordinate. The angle is measured from the positive x-axis. This calculator takes the angle in degrees, converts it to radians internally, and returns the exact x and y values of the point.
What are polar coordinates?
Polar coordinates locate a point using two values: a radius, the straight-line distance from the origin, and an angle, the direction from the positive x-axis. They contrast with Cartesian coordinates, which use horizontal and vertical distances. Polar coordinates are especially convenient for circular, rotational, or angular situations where distance and direction are the natural quantities.
Why convert between coordinate systems?
Different problems are easier in different systems. Polar coordinates simplify circular motion, navigation, and radar, while Cartesian coordinates are needed for plotting on a grid, computer graphics, and many algebraic calculations. Converting lets you work in whichever system suits the task and then translate results into the form you need, a common step in math, physics, and engineering.
Should the angle be in degrees or radians?
This calculator uses degrees for input because they are intuitive for most people, then converts to radians internally since the sine and cosine functions operate in radians. If your angle is given in radians, convert it to degrees first, or remember that 180 degrees equals pi radians. Using the wrong unit is a common source of errors in coordinate conversion.
Can the radius be negative?
By convention, a negative radius is allowed and means the point lies in the opposite direction from the given angle, equivalent to adding 180 degrees to the angle with a positive radius. The calculator handles negative radii correctly, producing x and y values that reflect this reversed direction. Most everyday uses, however, involve a positive radius.