Case 1
- Which way are you converting?
- Polar (r, θ) to rectangular (x, y)
- Radius r
- 5
- Angle θ (in the unit selected below)
- 53.13
- x coordinate
- 3
- y coordinate
- 4
- Angle unit
- Degrees
(3.000007, 3.999995)
Open with these valuesPolar to Rectangular Converter
(3.000007, 3.999995)
Result: (3.000007, 3.999995)Polar to rectangular: x = r × cos θ and y = r × sin θ. Rectangular to polar: r = √(x² + y²) and θ = atan2(y, x), normalised into 0–360°. The point (3, 4) sits 5 units from the origin at 53.13° — the 3-4-5 triangle in polar form.
Worked examples
Case 2
- Which way are you converting?
- Rectangular (x, y) to polar (r, θ)
- Radius r
- 5
- Angle θ (in the unit selected below)
- 0
- x coordinate
- -3
- y coordinate
- 4
- Angle unit
- Degrees
r = 5, θ = 126.869898°
Open with these valuesCase 3
- Which way are you converting?
- Rectangular (x, y) to polar (r, θ)
- Radius r
- 5
- Angle θ (in the unit selected below)
- 0
- x coordinate
- 0
- y coordinate
- -5
- Angle unit
- Degrees
r = 5, θ = 270°
Open with these valuesHow it's calculated
x = r cos θ, y = r sin θ; r = √(x² + y²), θ = atan2(y, x)
- StepPick the direction: polar to rectangular, or rectangular to polar.
- StepPick the angle unit — it applies to the angle you enter and the one you get.
- StepFill in the two values that direction needs: r and θ, or x and y.
- ResultRead the converted point; angles come back normalised into 0–360°.
What this number means
Rectangular (Cartesian) coordinates locate a point by how far across and how far up: (x, y). Polar coordinates locate the same point by how far away and in what direction: a radius r and an angle θ measured anticlockwise from the positive x-axis. The angle unit is yours to pick, and it applies to both the angle you type in and the one you get back — mixing degrees and radians silently is the most common source of a wrong answer here. Going from polar to rectangular is straightforward trigonometry: the radius is the hypotenuse and the coordinates are its projections, x = r × cos θ and y = r × sin θ. The reverse direction is where implementations go wrong. The radius is easy — it is Pythagoras, r = √(x² + y²) — but the angle needs more care than arctan(y ÷ x) gives. A plain arctangent returns an angle between −90° and 90°, so it cannot tell (3, 4) from (−3, −4), and at x = 0 the division fails outright. atan2 takes the signs of x and y separately, returns the correct quadrant and stays defined on the axes: (−3, 4) gives 126.87°, and (0, −5) gives 270°. This converter also normalises the result into 0–360°, so an angle never comes back negative. At the origin the angle is undefined — r = 0 describes that point whatever θ says.
Both directions, one page
Polar to rectangular and back, in the angle unit you choose. The unit applies to the angle you enter and the angle you get, so the two cannot get out of step.
arctan(y ÷ x) is not enough
It loses two quadrants and is undefined at x = 0. This converter uses atan2 and normalises the angle into 0–360°.
At the origin the angle is undefined
r = 0 describes the origin regardless of θ, so a converted angle there carries no information. Values around 10⁻¹⁶ where you expect zero are floating-point noise, not an error.
Commonly misread
The angle is just arctan(y ÷ x).
That only holds in the first and fourth quadrant. atan2(y, x) reads both signs and stays defined at x = 0.
A point below the x-axis should give a negative angle.
The result is normalised into 0–360°, so (0, −5) reads 270° rather than −90°. Both describe the same direction.
This angle is a compass bearing.
Bearings run clockwise from north; this angle runs anticlockwise from the positive x-axis. Convert with bearing = (90° − θ) mod 360°.
Reference table
| Direction | Input | Result |
|---|---|---|
| Polar → rectangular | r = 5, θ = 53.13° | (3.000007, 3.999995) |
| Rectangular → polar | (3, 4) | r = 5, θ = 53.130102° |
| Rectangular → polar | (−3, 4) | r = 5, θ = 126.869898° |
| Rectangular → polar | (0, −5) | r = 5, θ = 270° |
| Polar → rectangular | r = 2, θ = 1.570796 rad | (0, 2) |
| Rectangular → polar | (1, 1) | r = 1.414214, θ = 0.785398 rad |
Questions
What are the conversion formulas?
Polar to rectangular: x = r × cos θ and y = r × sin θ. Rectangular to polar: r = √(x² + y²) and θ = atan2(y, x). For the 3-4-5 triangle, (3, 4) becomes r = 5 at 53.13°, and back again.
Why atan2 instead of arctan(y/x)?
A plain arctangent returns angles between −90° and 90° only, so (3, 4) and (−3, −4) come out identical, and at x = 0 the division fails outright. atan2 reads the signs of x and y separately, so (−3, 4) correctly gives 126.87° and (0, −5) gives 270°.
Why is my angle never negative?
Because the result is normalised into 0–360°. atan2 natively returns −180° to 180°, so a point below the x-axis would come back as a negative angle. Adding a full turn gives the same direction in the conventional range — (0, −5) reads 270° rather than −90°.
Should I use degrees or radians?
Radians for calculus and programming, degrees for engineering and surveying. The unit selector applies to both the angle you type in and the one you get back, so the two cannot get out of step. Mixing them silently is the most common cause of a wrong answer in coordinate work.
Why do I get 1.9e-16 instead of 0?
Floating-point rounding, not an error. Converting r = 10 at exactly 90° should give x = 0, but π ÷ 2 cannot be represented exactly in binary, so the cosine comes out as a number around 10⁻¹⁶ instead. Anything at that magnitude is zero for every practical purpose.
Does this match compass bearings?
No — the conventions differ. This calculator uses the mathematical convention: anticlockwise from the positive x-axis, which points east. Navigation bearings are measured clockwise from north, so convert with bearing = (90° − θ) mod 360°.
Sources and last check
Information, not professional advice.