How this instrument works
Polar coordinates describe a point by its distance r from an origin and its angle θ from a reference direction — a natural way to describe position when something is naturally tracked by direction and range, like a radar contact, a robot arm's reach, or a compass bearing and distance. Rectangular (Cartesian) coordinates describe the same point instead by how far it sits along two perpendicular axes, x and y. Converting from polar to rectangular applies x = r·cos(θ) and y = r·sin(θ), the same relationship that places any point on a circle of radius r.
This conversion is essential whenever a system built around plain (x, y) coordinates — a map, a game engine, a plotting library — needs to work with data that arrived in polar form instead. Sensor readings, GPS bearings, and physics problems phrased in terms of distance-and-direction all eventually need this translation before ordinary Cartesian-based tools and formulas can use them.
The math here is identical to converting a complex number from trigonometric (polar) form to rectangular form elsewhere on this site — the same r and θ, the same cos and sin — but framed for physical position in space rather than for a complex number's real and imaginary parts. Same relationship, different practical context.
- Enter the point's distance from the origin into the Radial distance (r) field.
- Enter the point's angle from the reference direction into the Angle (θ) field.
- Read x and y: the sheet applies r·cos(θ) and r·sin(θ) respectively.
Worked example — distance 5, angle 53.13°
A point sits at polar distance 5, angle 53.13° from the reference direction. Converting to rectangular coordinates gives x = 5×cos(53.13°) ≈ 3, y = 5×sin(53.13°) ≈ 4 — landing on the point (3, 4), the familiar 3-4-5 triangle appearing again as a position rather than as a triangle's own sides.
A point 10 units out at an angle of exactly 0° converts to (10, 0) — sitting directly along the reference direction with no perpendicular offset. A point 4 units out at 90° converts to (0, 4) instead — a quarter turn sends the point entirely along the perpendicular axis, with nothing left along the original reference direction.
Questions
How do you convert polar coordinates to rectangular coordinates?
x = r·cos(θ) and y = r·sin(θ), where r is the distance from the origin and θ is the angle from the reference direction. This is the same relationship that places any point on a circle of radius r at angle θ.
When are polar coordinates used instead of rectangular ones?
Whenever a position is naturally described by distance and direction rather than by two perpendicular offsets — radar tracking, compass bearings, robotic arm reach, and orbital mechanics all commonly use polar-style coordinates as their natural starting description.
Is this the same as converting a complex number's polar form?
Mathematically identical — both use the same x = r·cos(θ), y = r·sin(θ) relationship. This page frames it for physical position in space; the companion complex-number page frames the identical math for a complex number's real and imaginary parts.
What happens at an angle of 0°?
The y-coordinate becomes exactly zero, since sin(0°) = 0, and the x-coordinate equals the full radial distance, since cos(0°) = 1 — the point sits entirely along the reference direction, with no perpendicular offset at all.
Can the radial distance be negative?
By convention, r is usually taken as non-negative, with any 'backward' direction instead represented by adjusting the angle θ by 180° — this calculator expects a non-negative r for that reason, matching standard polar-coordinate convention.