Two ways to locate a point
A point in the plane can be described by its rectangular (Cartesian) coordinates, x across and y up, or by its polar coordinates: the distance r from the origin and the angle θ measured counterclockwise from the positive x-axis. Polar coordinates are natural for anything that turns or radiates: radar screens, circular motion, antennas, complex numbers and spirals.
The formulas
- To polar: r = √(x² + y²) and θ = atan2(y, x)
- To rectangular: x = r cos θ and y = r sin θ
- Degrees to radians: multiply by π ÷ 180
The function atan2 returns an angle between −180° and 180° in the correct quadrant. The calculator also gives the same angle between 0° and 360°, the convention used in navigation and many textbooks.
Worked example
Convert the point (−3, 4) to polar coordinates.
- r = √(9 + 16) = 5.
- A plain arctan(4 ÷ −3) would give −53.13°, which points into the fourth quadrant. Since x is negative and y positive, the point lies in the second quadrant: θ = −53.13° + 180° = 126.87°.
- In radians: 126.87 × π ÷ 180 = 2.2143 rad.
Going the other way, r = 5 and θ = 30° give x = 5 cos 30° = 4.330 and y = 5 sin 30° = 2.5.
Quadrants and angles
| Quadrant | Signs | Angle range |
|---|---|---|
| I | x > 0, y > 0 | 0° to 90° |
| II | x < 0, y > 0 | 90° to 180° |
| III | x < 0, y < 0 | 180° to 270° (or −180° to −90°) |
| IV | x > 0, y < 0 | 270° to 360° (or −90° to 0°) |
Points on the axes have angles of 0°, 90°, 180° or 270°. At the origin, r = 0 and the angle is undefined; the calculator shows 0°.
Common uses
- Physics: circular motion and rotating objects are described by a radius and an angle that changes with time.
- Complex numbers: the polar form r(cos θ + i sin θ) makes multiplication easy, since the radii multiply and the angles add.
- Engineering: radar and sonar report targets as a range and a bearing.
- Graphics: polar curves such as r = 1 + cos θ draw cardioids and rose curves.
One point, many angles
The same point can be written with angles that differ by 360°: (5, 126.87°), (5, 486.87°) and (5, −233.13°) are identical. Some conventions also allow a negative radius, pointing in the opposite direction. The calculator always returns r ≥ 0 and the principal angle.