Last updated: 2026-07-01

Polar and rectangular coordinates

A point in the plane can be located in two equivalent ways. Rectangular (Cartesian) coordinates (x, y) give the horizontal and vertical distance from the origin. Polar coordinates (r, θ) give the straight-line distance r from the origin and the angle θ that the line to the point makes with the positive x-axis. This converter moves between the two systems in either direction, with the angle expressed in degrees or radians.

Both descriptions point to the same place; which one is more convenient depends on the problem. Rectangular coordinates suit grids, screens, and component-wise arithmetic, while polar coordinates suit rotations, circular motion, and anything described by a distance and a heading.

Polar to rectangular

Drop a right triangle from the point to the x-axis. The radius r is the hypotenuse, the angle θ sits at the origin, and the two legs are exactly the coordinates:

• x = r·cos θ
• y = r·sin θ

The cosine gives the horizontal leg and the sine gives the vertical leg, so the signs of x and y follow automatically from the quadrant of θ.

Rectangular to polar

Going the other way reverses the triangle. The radius is the hypotenuse, recovered with the Pythagorean theorem, and the angle is the direction of the point:

• r = √(x² + y²)
• θ = atan2(y, x)

The two-argument arctangent atan2(y, x) is used instead of arctan(y/x) because the plain ratio y/x cannot tell a point from its opposite — both give the same quotient. By keeping x and y separate, atan2 reads their signs and returns the angle in the correct quadrant, across the full range from −180° to 180° (or −π to π in radians). It is also defined when x = 0, where arctan(y/x) would divide by zero.

Worked example

Problem: Convert the rectangular point (3, 4) to polar form.

The radius is the distance from the origin:

r = √(3² + 4²) = √(9 + 16) = √25 = 5

The angle is the direction of the point:

θ = atan2(4, 3) ≈ 53.13°

So (3, 4) becomes (5, 53.13°). Converting back confirms the round trip: x = 5·cos 53.13° ≈ 3 and y = 5·sin 53.13° ≈ 4.

The origin and the angle range

The origin is the one point where polar form is ambiguous: r = 0 leaves θ undefined, since every direction lands on the same point. Away from the origin, the angle returned by atan2 lies between −180° and 180°. Other conventions exist — some texts report angles in [0°, 360°) — so adding 360° (or 2π) to a negative angle gives the equivalent positive value when that range is preferred.

Choosing degrees or radians

Degrees and radians measure the same angles: a full turn is 360° or 2π radians, and 180° = π radians. Degrees are the norm in geometry, surveying, and navigation; radians are the standard in calculus and physics because they make derivatives of the trigonometric functions clean. Set the angle unit to match your source values, and the converter takes care of the internal conversion in both directions.