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Polar to Cartesian Converter

Cartesian x and y for a polar point, with the angle measured counter-clockwise.

How it is calculated

A point given by distance r from the origin and angle θ has Cartesian coordinates x = r × cos(θ) and y = r × sin(θ). The angle must be entered in degrees, measured counter-clockwise from the positive x axis; the tool converts it to radians (θ × π ÷ 180) before applying cos and sin. There is no limit on the angle, so 390° gives the same point as 30°, and a negative angle turns clockwise. A negative r is accepted and places the point on the opposite side of the origin, equal to adding 180° to the angle. Both results are rounded to two decimals, so tiny values such as the x of a 90° angle are shown as 0.00.

Worked example

Take r = 10 and θ = 30°. cos 30° ≈ 0.8660 and sin 30° = 0.5. So x = 10 × 0.8660 = 8.66 and y = 10 × 0.5 = 5.00. The point is about 8.66 units to the right of the origin and 5 units up, which lies on the 30° line at distance 10, since √(8.66² + 5²) ≈ 10.

Try this example →

Common mistakes

  • Entering the angle in radians when the field takes degrees. Typing 0.52 for 30° gives a point almost on the x axis.
  • Measuring the angle from the y axis or clockwise, as a compass bearing does. Here 0° points along +x and angles increase counter-clockwise, so a compass bearing of 30° must be converted first (90° − bearing).
  • Swapping x and y or the cos/sin roles. The x value uses cosine and the y value uses sine, not the other way round.
  • Expecting a negative radius to be rejected. It is accepted and flips the point to the opposite side, which is easy to do by accident.
  • Reading 0.00 as exact. Results are rounded to two decimals, so a very small x or y shows as zero.

About this calculator

Polar coordinates locate a point by a radius r (distance from the origin) and an angle θ measured from a reference axis, while Cartesian coordinates locate the same point with horizontal and vertical distances x and y. The conversion follows directly from basic trigonometry: x = r·cos(θ) and y = r·sin(θ).

This conversion is routine in physics and engineering, where forces, velocities and electromagnetic fields are often measured as a magnitude and direction but need to be added or plotted component-wise. Robotics and navigation systems use it to turn a radar or sonar reading (distance and bearing to a target) into x/y coordinates for mapping, and graphics programmers use it to place points on circular or spiral paths.

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Comments (1)

  • Editorial team

    A quick sanity check is to square both answers and add them: x squared plus y squared must equal r squared. The default of r = 5 at 53.13 degrees gives x = 3.00 and y = 4.00, and 9 plus 16 is 25. The coordinates are shown rounded to two decimals, so converting them back by hand can differ slightly from your starting angle. Do you also need the reverse direction, or a third axis for 3D points?

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