What are the polar coordinates?
Type a point as (x, y) to get its polar coordinates, or as (r, θ) to get x and y. The polar coordinates calculator shows the angle in degrees and radians and draws the point on the plane.
- Converted point
- (r, θ) = (4.242640687, 45°)
The point converts to (r, θ) = (4.242640687, 45°).
- x
- 3
- y
- 3
- r
- 4.242641
- θ in degrees
- 45
- θ in radians
- 0.785398
Converted point: (r, θ) = (4.242640687, 45°). The point converts to (r, θ) = (4.242640687, 45°).
Where is the point?
How to calculate
Converts a point between rectangular coordinates (x, y) and polar coordinates (r, θ), with θ in degrees and radians, and draws the point.
Example with the default inputs (Convert (x, y) to polar, x 3, y 3): The point converts to (r, θ) = (4.242640687, 45°).
Method: Rectangular to polar: r = √(x² + y²), θ = atan2(y, x), plus 360° when negative. Polar to rectangular: x = r cos θ, y = r sin θ.
- θ is measured counterclockwise from the positive x-axis. The page gives θ from 0° up to 360°; adding or taking away any whole number of turns (360° or 2π) names the same point.
- A typed r may be negative: the point is then on the opposite side of the origin, as in OpenStax.
Worked examples
Each example is checked against the calculator on every build.
- Convert (x, y) to polar, x 3, y 3 gives r 4.242641, θ in degrees 45, θ in radians 0.785398, Converted point (r, θ) = (4.242640687, 45°).Source: OpenStax, Algebra and Trigonometry 2e, §10.3 Polar Coordinates, https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-3-polar-coordinates (Example 5: r = 3√2, θ = π/4)
- Convert (r, θ) to rectangular, r 3, θ 90, Angle unit Degrees gives x 0, y 3, Converted point (x, y) = (0, 3).Source: OpenStax, Algebra and Trigonometry 2e, §10.3 Polar Coordinates, https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-3-polar-coordinates (Example 3: (3, π/2) is (0, 3))
- Convert (r, θ) to rectangular, r -2, θ 0, Angle unit Radians gives x -2, y 0.Source: OpenStax, Algebra and Trigonometry 2e, §10.3 Polar Coordinates, https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-3-polar-coordinates (Example 4: (−2, 0) is (−2, 0))
- Convert (x, y) to polar, x -1, y -1.732051 gives r 2, θ in degrees 240, θ in radians 4.18879.Source: OpenStax, Algebra and Trigonometry 2e, §10.3 Polar Coordinates, https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-3-polar-coordinates
- Convert (r, θ) to rectangular, r 2, θ 30, Angle unit Degrees gives x 1.732051, y 1.Source: OpenStax, Algebra and Trigonometry 2e, §10.3 Polar Coordinates, https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-3-polar-coordinates
How it works
Rectangular to polar. From a point (x, y):
- r = √(x² + y²), the distance from the origin (computed with a guarded hypotenuse function, so very large or small values do not overflow).
- θ = atan2(y, x): the angle from the positive x-axis to the point, which uses the signs of x and y to pick the quadrant. It comes out from −180° to 180°; a negative angle has 360° (2π) added, so θ is from 0° up to 360°.
- θ in degrees = θ in radians × 180 ÷ π.
- At the origin (0, 0) there is no angle, so the page gives no answer.
Polar to rectangular. From (r, θ), with θ typed in degrees (changed to radians as θ × π ÷ 180) or in radians:
- x = r cos θ and y = r sin θ.
- A coordinate within 10⁻¹² × |r| of 0 shows as 0, because floating-point cos 90° is 6.1 × 10⁻¹⁷, not 0.
Rules. Every typed number is from −1,000,000,000 to 1,000,000,000.
Output format. The converted point is text, each number with up to 10 significant figures and no thousands separators. The separate x, y, r and θ values show in the site's number format.
Worked examples by hand
(3, 3) to polar (OpenStax Example 5). r = √(9 + 9) = √18 = 3√2 = 4.2426. θ = atan2(3, 3) = π/4 = 45° (0.7854 rad).
(3, 90°) to rectangular (OpenStax Example 3). x = 3 cos 90° = 0; y = 3 sin 90° = 3.
(−2, 0) to rectangular (OpenStax Example 4). x = −2 cos 0 = −2; y = −2 sin 0 = 0.
(−1, −√3) to polar. r = √(1 + 3) = 2. atan2(−√3, −1) = −120°; + 360° = 240° = 4π/3 = 4.1888 rad.
(2, 30°) to rectangular. x = 2 cos 30° = √3 = 1.7321; y = 2 sin 30° = 1.
Other questions people ask
How do I convert rectangular coordinates to polar?
Find the distance r = √(x² + y²) and the angle θ whose tangent is y ÷ x, in the right quadrant. For (3, 3), r = √18 = 4.2426 and θ = 45° (π/4 radians).
How do I convert polar coordinates to rectangular?
Multiply r by the cosine and the sine of the angle: x = r cos θ and y = r sin θ. The point (3, 90°) is x = 3 × 0 = 0, y = 3 × 1 = 3, so (0, 3).
Why is my angle 240° and not 60°?
tan θ = y ÷ x is the same for a point and for the point opposite it through the origin. The page uses atan2, which looks at the signs of x and y, so (−1, −√3) gets 240° (third quadrant), not 60°.
What range does θ use?
The page gives θ from 0° up to, but not including, 360° (0 to 2π radians). The same point also has angles θ ± 360°, and θ ± 180° with a negative r.
Can r be negative?
Yes, when you convert from polar. A negative r points in the direction opposite to θ, so (−2, 0) is the point (−2, 0). When you convert to polar the page always gives r ≥ 0.
What about the origin?
At (0, 0) the distance r is 0 and every angle names the same point, so θ has no single value. The page asks you to move the point away from the origin.