Polar to Cartesian Converter: x and y from r and θ

How it works

Enter the values

Type the numbers or the formula. Decimals may use a point or a comma. No idea? Click Fill in an example.

Instant answer

The answer appears as you type, with the most important intermediate values.

See the working

Under "How it's worked out" you see the steps, handy for checking your own calculation.

The formulas

x = r · cos(θ)
y = r · sin(θ)

For r = 5 and θ = 53.13°: x ≈ 3 and y ≈ 4. The angle may be in degrees or radians; choose the unit above the fields.

The other way round is the Cartesian to polar converter.

Worked examples

r	θ	(x, y)
2	60°	(1, 1.7321)
5	53.13°	(3, 4)
10	90°	(0, 10)
4	180°	(−4, 0)

From a compass bearing

A boat sails 100 m on a bearing of 30° (clockwise from north). The mathematical angle is 90° − 30° = 60°, so x = 100 × cos 60° = 50 m east and y = 100 × sin 60° ≈ 86.6 m north.

Programming and graphics

Game objects that orbit, radar plots and polar charts all store a radius and an angle and convert to (x, y) to draw. The same idea gives the position of a point on a circle: (r cos θ, r sin θ). To convert in the other direction, use the Cartesian to polar converter; for complex numbers in polar form, see multiply complex numbers.

Frequently asked questions

Is the angle measured from the x-axis?

Yes, counterclockwise from the positive x-axis. A compass bearing is measured clockwise from north, so convert it first: θ = 90° − bearing.

Can r be negative?

Yes. A negative r points in the opposite direction, so (−2, 60°) lands at the same place as (2, 240°).

Why does cos 90° give exactly 0 here?

Computers leave a tiny remainder, such as 6 × 10⁻¹⁷, for cos(π/2). The calculator rounds results like that to 0, so r = 10 at 90° shows (0, 10).

Can I enter the angle as pi/3?

Yes, in radians mode sums like pi/3 and 2pi/5 are accepted.