Math
Polar Coordinates Calculator
Enter a point as (x, y) or as (r, θ) and mathcheck converts it the other way.
Rectangular to polar gives r = sqrt(x^2 + y^2) as an exact radical plus θ from a reference angle with the quadrant rule written out, in both degrees and radians. Polar to rectangular accepts degrees, radians, or π fractions like 2pi/3 and returns exact values such as (-2, 2√3) whenever θ is a special angle. The plot shows the point on a polar grid with r and θ marked.
Horizontal coordinate of the point.
Vertical coordinate of the point.
Try an example
Result
Converted point
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- r
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- θ (degrees)
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- θ (radians)
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- x
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- y
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- Quadrant
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More details (1 more)
- Other polar names
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Study path
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What polar coordinates are
Rectangular coordinates locate a point by how far it is right (x) and up (y). Polar coordinates locate the same point by its distance from the origin (r) and the angle θ its direction makes with the positive x-axis, measured counterclockwise. Both describe the same point; the conversions are just right-triangle trigonometry.
Getting the angle right in every quadrant
A plain arctangent only returns angles between -90° and 90°, so it cannot tell (-1, -1) apart from (1, 1). Find the acute The small positive angle between a given angle's arm and the x-axis. α = atan(|y|/|x|) first, then place it using the signs of x and y.
| Signs | Quadrant | θ in degrees | θ in radians |
|---|---|---|---|
| x > 0, y > 0 | I | α | α |
| x < 0, y > 0 | II | 180° - α | π - α |
| x < 0, y < 0 | III | 180° + α | π + α |
| x > 0, y < 0 | IV | 360° - α (or -α) | 2π - α (or -α) |
| x = 0 | on the y-axis | 90° or 270° | π/2 or 3π/2 |
| y = 0 | on the x-axis | 0° or 180° | 0 or π |
(-1, 1) to polar
r = sqrt(1 + 1) = √2. Reference angle α = atan(1/1) = 45°. The point is in Quadrant II, so θ = 180° - 45° = 135° = 3π/4. Polar form: (√2, 135°).
(4, 150°) to rectangular
cos 150° = -√3/2 and sin 150° = 1/2, so x = 4(-√3/2) = -2√3 ≈ -3.464 and y = 4(1/2) = 2. The point is (-2√3, 2).
One point, many polar names
Unlike rectangular coordinates, polar coordinates are not unique. Adding or subtracting 360° (2π) to θ gives the same point, and so does flipping the sign of r while adding 180° (π): (r, θ) = (r, θ + 360°) = (-r, θ + 180°). The calculator lists these alternate names so you can match whatever form your textbook uses.
How to use this calculator
- Pick the direction: rectangular to polar or polar to rectangular.
- For rectangular to polar, type x and y. For polar to rectangular, type r and θ, choosing degrees or An angle unit where a full turn is 2π instead of 360 degrees. (π fractions like 5pi/6 are always radians).
- Read the converted point in the header, then r, θ in degrees and radians, x, y, and the One of the four regions the x- and y-axes cut the coordinate plane into. in the grid.
- Check the plot to confirm the point sits where you expect on the polar grid.
- Open Show the work to see the A root expression written with the √ symbol, such as a square root. for r, the reference angle, the quadrant rule, and the exact sine and cosine values.
How to read the answer
r is the straight-line distance from the origin and is never negative in the rectangular-to-polar result. θ is reported in [0°, 360°) with the negative equivalent in parentheses when the point is below the x-axis, because both are common. In the polar-to-rectangular result, a value like (-2√3, 2) is exact; the decimal form follows in the header when the The answer kept as a fraction or radical instead of a rounded decimal. contains a radical.
Common mistakes and edge cases
- Using atan(y/x) without checking the quadrant: (-1, -1) gives 45° from the calculator button, but the true angle is 225°.
- Mixing degrees and radians. If a problem says θ = 2, it means 2 radians (about 114.6°), not 2°.
- Forgetting that r can be negative in polar form: (-3, 60°) is the same point as (3, 240°).
- Treating polar coordinates as unique. Answers that differ by 360° in θ, or by a sign flip in r with a 180° shift, are the same point.
- Dividing by x when x = 0. Points on the y-axis have θ = 90° or 270° directly.
Worked examples
Quadrant II point to polar
(-1, 1)
Converted point
(√2, 135°) = (√2, 3π/4)
3-4-5 point to polar
(3, 4)
Converted point
(5, 53.1301°)
Quadrant III with a special angle
(-1, -√3) entered as decimals
Converted point
(2, 240°) = (2, 4π/3)
Point on the negative y-axis
(0, -5)
Converted point
(5, 270°) = (5, 3π/2)
The origin
(0, 0) has r = 0 and no direction
Converted point
(0, 0°) = (0, 0)
Polar to rectangular, exact
(4, 150°)
Converted point
(-2√3, 2) ≈ (-3.4641, 2)
Polar to rectangular with a π fraction
(2, 5π/6)
Converted point
(-√3, 1) ≈ (-1.7321, 1)
Negative radius
(-3, 60°) is the same point as (3, 240°)
Converted point
(-3/2, -3√3/2) ≈ (-1.5, -2.5981)
Non-special angle in radians
(2, 1 rad)
Converted point
(1.0806, 1.6829)
Unreadable angle
Converted point
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Frequently asked questions
How do I convert rectangular coordinates to polar?+
Compute r = sqrt(x^2 + y^2). Then find the reference angle α = atan(|y|/|x|) and adjust for the quadrant: θ = α in Quadrant I, 180° - α in II, 180° + α in III, and 360° - α in IV. Points on an axis use 0°, 90°, 180°, or 270° directly.
How do I convert polar coordinates to rectangular?+
Use x = r cos θ and y = r sin θ. For special angles use exact values, for example cos 150° = -√3/2, so (4, 150°) becomes (-2√3, 2).
Is θ in degrees or radians?+
Either, as long as you are consistent. This calculator shows both and lets you type radians as a π fraction such as 2pi/3. Calculus and physics usually use radians; many geometry courses use degrees.
Can r be negative?+
Yes. A negative r means walk backwards along the ray: (-r, θ) is the same point as (r, θ + 180°). When converting from rectangular coordinates the calculator always reports the standard form with r ≥ 0.
Why does my calculator give the wrong angle for (-2, -2)?+
The inverse-tangent button returns only angles between -90° and 90°. atan(-2/-2) = atan(1) = 45°, but the point is in Quadrant III, so the correct angle is 225°. Always check the signs of x and y.
What is the polar form of the origin?+
r = 0 with any angle. The origin (called the pole) is the only point without a unique direction, so (0, θ) names it for every θ.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026