Polar Cartesian Converter
Use this Polar Cartesian Converter to enter values, adjust options, and review results in a compact responsive workspace.
Results are calculated automatically as you enter data.
Converted coordinates
Formulas
x = r cos(theta), y = r sin(theta)
r = sqrt(x^2 + y^2), theta = atan2(y, x)
Interactive graphs
Drag either point to update both systems.▼ See explanations and tips below ▼
Related Calculators
What Are Polar and Cartesian Coordinates?
Cartesian and polar coordinates are two ways to describe the same point in a plane.
In Cartesian form, a point is written as \((x, y)\). The value \(x\) tells you how far the point is horizontally from the origin, and \(y\) tells you how far it is vertically from the origin. Positive \(x\) moves right, negative \(x\) moves left, positive \(y\) moves up, and negative \(y\) moves down.
In polar form, a point is written as \((r, \theta)\). The value \(r\) is the distance from the origin to the point. The angle \(\theta\) tells you the direction of that distance, usually measured from the positive \(x\)-axis.
Both forms describe location, but they emphasize different ideas. Cartesian coordinates describe a point by its horizontal and vertical components. Polar coordinates describe a point by its distance and direction.
Why Coordinate Conversion Matters
Coordinate conversion is useful because some problems are easier to understand in one coordinate system than the other.
Cartesian coordinates are convenient for straight-line movement, grid locations, graphing functions such as \(y = mx + b\), and comparing horizontal and vertical distances. Polar coordinates are useful when a situation naturally involves direction, rotation, circular motion, distance from a center, or vectors.
For example, a point that is 5 units from the origin at an angle of \(45^\circ\) is simple to describe in polar form. But if you need to plot it on a rectangular grid, you need its \(x\) and \(y\) values. Converting between the two forms lets you move between geometric meaning and graph-ready coordinates.
Key Terms to Know
- Origin: The point \((0, 0)\), where the \(x\)-axis and \(y\)-axis meet.
- Cartesian coordinates: The ordered pair \((x, y)\), describing horizontal and vertical position.
- Polar coordinates: The ordered pair \((r, \theta)\), describing distance from the origin and angle from the positive \(x\)-axis.
- Radius or radial coordinate: The value \(r\), which represents distance from the origin.
- Angle or angular coordinate: The value \(\theta\), which represents direction.
- Radians: An angle measure based on arc length. A full turn is \(2\pi\) radians.
- Degrees: An angle measure where a full turn is \(360^\circ\).
- Quadrant: One of the four regions of the coordinate plane. Quadrants matter when finding the correct angle.
- Angle normalization: Rewriting an angle into a chosen standard range, such as \(0 \le \theta < 2\pi\) or \(-\pi < \theta \le \pi\).
How Coordinate Conversion Works
Polar and Cartesian conversion comes from right-triangle trigonometry. If a point has polar coordinates \((r, \theta)\), then \(r\) acts like the hypotenuse of a right triangle, \(x\) is the horizontal side, and \(y\) is the vertical side.
To convert from polar to Cartesian form, use cosine for the horizontal component and sine for the vertical component:
$$
x = r\cos(\theta)
$$
$$
y = r\sin(\theta)
$$
Where:
- \(x\) is the horizontal Cartesian coordinate.
- \(y\) is the vertical Cartesian coordinate.
- \(r\) is the distance from the origin.
- \(\theta\) is the angle measured from the positive \(x\)-axis.
To convert from Cartesian to polar form, use the Pythagorean theorem to find the distance from the origin:
$$
r = \sqrt{x^2 + y^2}
$$
Then find the angle:
$$
\theta = \operatorname{atan2}(y, x)
$$
The two-input arctangent, usually written as \(\operatorname{atan2}(y, x)\), is preferred because it uses the signs of both \(x\) and \(y\) to identify the correct quadrant. A simple \(\arctan\left(\frac{y}{x}\right)\) calculation can give the wrong direction when the point is in the left half of the coordinate plane or when \(x = 0\).
To convert between degrees and radians:
$$
\theta_{\text{rad}} = \theta_{\text{deg}} \times \frac{\pi}{180}
$$
$$
\theta_{\text{deg}} = \theta_{\text{rad}} \times \frac{180}{\pi}
$$
Examples of Coordinate Conversion in Practice
Example 1: Converting \((3, 4)\) from Cartesian to Polar Form
Start with the Cartesian point:
$$
(x, y) = (3, 4)
$$
Find the radius:
$$
r = \sqrt{3^2 + 4^2}
$$
$$
r = \sqrt{9 + 16} = \sqrt{25} = 5
$$
Now find the angle:
$$
\theta = \operatorname{atan2}(4, 3)
$$
$$
\theta \approx 0.9273\text{ rad} \approx 53.1301^\circ
$$
So the point \((3, 4)\) is approximately:
$$
(5, 53.1301^\circ)
$$
or:
$$
(5, 0.9273\text{ rad})
$$
Example 2: Converting \((5, 45^\circ)\) from Polar to Cartesian Form
Start with:
$$
r = 5, \qquad \theta = 45^\circ
$$
Since \(45^\circ = \frac{\pi}{4}\), use the exact unit-circle values:
$$
\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}
$$
$$
\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}
$$
Find \(x\) and \(y\):
$$
x = 5\cos\left(\frac{\pi}{4}\right) = 5\cdot\frac{\sqrt{2}}{2} = \frac{5\sqrt{2}}{2}
$$
$$
y = 5\sin\left(\frac{\pi}{4}\right) = 5\cdot\frac{\sqrt{2}}{2} = \frac{5\sqrt{2}}{2}
$$
As decimals:
$$
x \approx 3.5355, \qquad y \approx 3.5355
$$
So the Cartesian form is approximately:
$$
(3.5355, 3.5355)
$$
Example 3: A Normalized Angle Such as \(450^\circ\)
Angles that differ by a full turn point in the same direction. Since a full turn is \(360^\circ\), the angle \(450^\circ\) points in the same direction as \(90^\circ\):
$$
450^\circ - 360^\circ = 90^\circ
$$
For \((r, \theta) = (2, 450^\circ)\), use the equivalent angle \(90^\circ\):
$$
x = 2\cos(90^\circ) = 0
$$
$$
y = 2\sin(90^\circ) = 2
$$
So \((2, 450^\circ)\) and \((2, 90^\circ)\) describe the same point:
$$
(0, 2)
$$
This is why a converter may rewrite large or negative angles into a standard range.
How to Interpret the Result
A converted coordinate result should be read as another description of the same point, not as a new point.
If you start with Cartesian coordinates, the converted polar result tells you:
- \(r\): how far the point is from the origin.
- \(\theta\): the direction from the positive \(x\)-axis to the point.
If you start with polar coordinates, the converted Cartesian result tells you:
- \(x\): how far the point is to the right or left of the origin.
- \(y\): how far the point is above or below the origin.
A larger \(r\) means the point is farther from the origin. A value of \(r = 0\) means the point is exactly at the origin, no matter what angle is written with it.
Angle format matters. A result shown in degrees and a result shown in radians can represent the same direction. For example:
$$
180^\circ = \pi\text{ rad}
$$
If the result includes an exact \(x/y\) form, it is a symbolic helper for common special angles such as \(0^\circ\), \(30^\circ\), \(45^\circ\), \(60^\circ\), \(90^\circ\), and their related quadrant angles. If it says there is no simple exact form, that does not mean the decimal result is wrong. It only means the angle was not one of the recognized special-angle cases.
Common Mistakes and Misconceptions
One common mistake is using the wrong angle unit. An angle of 45 means \(45^\circ\) when the angle unit is degrees, but it means 45 radians when the angle unit is radians. Those are very different directions.
Another mistake is expecting an angle such as \(450^\circ\) to remain unchanged. Since \(450^\circ\) and \(90^\circ\) point in the same direction, a converter may display the normalized angle instead of the original angle.
A third mistake is using ordinary arctangent without checking the quadrant. The ratio \(\frac{y}{x}\) can be the same for points in opposite quadrants, so the angle needs quadrant-aware handling.
Users also sometimes type unit labels directly into numeric fields, such as 5 cm or 45 deg. Coordinate converters usually expect only the number, with the unit or angle mode selected separately.
Finally, remember that a negative Cartesian coordinate is normal, but this calculator treats the polar radius as a nonnegative distance. In general mathematics, negative polar radii can be interpreted by pointing in the opposite direction, but this calculator uses \(r \ge 0\) for clarity.
When to Use Coordinate Conversion
Use coordinate conversion when you need to move between grid position and distance-direction form.
Common uses include:
- Plotting a polar point on a rectangular graph.
- Finding the magnitude and direction of a two-dimensional vector.
- Converting graphing problems between rectangular and polar form.
- Working with circular motion, rotation, or angles.
- Understanding special-angle values from the unit circle.
- Connecting rectangular and polar forms of complex numbers.
- Checking whether two differently written coordinate pairs describe the same point.
Limitations and Things to Keep in Mind
Coordinate conversion is exact in theory, but decimal calculator results are displayed with practical precision: up to 8 decimal places in the ordinary range and a 6-digit scientific-notation mantissa for nonzero values below \(10^{-6}\) or at least \(10^8\). Trailing zeros are removed. This is display rounding only: every finite nonzero coordinate, including a subnormal value such as \(5\times10^{-324}\), is retained rather than being replaced by zero.
Finite coordinates and radii are still converted when their magnitude is above \(10^{12}\), but the Cartesian and polar graphs are intentionally replaced with a scale-limit notice. This keeps extreme inputs from producing an unreadable or unsafe visualization.
The calculator changes signed negative zero to 0 and can snap a value that is extremely close to a nonzero integer to that integer. It does not use an absolute tiny-value cutoff. Degree inputs are reduced modulo \(360^\circ\) before conversion to radians, and radian inputs are reduced modulo \(2\pi\); this keeps huge but finite angles usable. The signed range is \(-\pi < \theta \le \pi\) and the unsigned range is \(0 \le \theta < 2\pi\). If Cartesian inputs make \(\sqrt{x^2+y^2}\) too large to represent as a finite number, the calculator reports a radius-overflow error instead of showing a successful undefined result.
Polar coordinates are not always unique. For example, adding a full turn to the angle gives the same point:
$$
(r, \theta) = (r, \theta + 2\pi)
$$
or, in degrees:
$$
(r, \theta) = (r, \theta + 360^\circ)
$$
This calculator may normalize the angle to either a signed range, such as \(-\pi < \theta \le \pi\), or an unsigned range, such as \(0 \le \theta < 2\pi\), depending on the selected angle range.
At the origin, the angle is not geometrically meaningful. The polar coordinates \((0, 0^\circ)\), \((0, 90^\circ)\), and \((0, \pi)\) all describe the same point: the origin. When a zero-radius point is synchronized through the calculator, the displayed angle may not preserve the angle that was originally typed.
The exact symbolic helper is limited to common special angles. It is not a full algebra system, and it will not simplify every possible trigonometric value.
This calculator is for two-dimensional Cartesian and polar coordinates. It does not convert three-dimensional cylindrical or spherical coordinates, and it does not parse real-world unit labels such as meters, feet, or centimeters.
How to Use This Calculator
- Choose Cartesian mode if you want to start with \(x\) and \(y\).
- Choose polar mode if you want to start with radius \(r\) and angle \(\theta\).
- Enter the required numeric values for the active mode.
- For polar input, choose whether the angle is in degrees or radians.
- Choose the angle range if signed or unsigned normalization is available.
- Review the main conversion result and the supporting Cartesian, polar, radians, and exact-value displays.
- Drag the graph point, if available, to explore how the coordinate values change visually.
- Use copy or graph download options, if available, when you need to save or reuse the result.
Frequently Asked Questions
What is the difference between Cartesian and polar coordinates?
Cartesian coordinates describe a point using horizontal and vertical movement: \((x, y)\). Polar coordinates describe the same kind of point using distance and direction: \((r, \theta)\). The two forms are different descriptions of location in the same plane.
Why does the angle sometimes change after conversion?
Angles that differ by a full turn point in the same direction, so a converter may normalize the angle into a standard range. For example, \(450^\circ\) may be displayed as \(90^\circ\) because they represent the same direction.
Why use \(\operatorname{atan2}(y, x)\) instead of \(\arctan\left(\frac{y}{x}\right)\)?
The two-input arctangent uses the signs of both \(x\) and \(y\), so it can identify the correct quadrant. The one-input arctangent only sees the ratio \(\frac{y}{x}\), which can be ambiguous for points in different quadrants.
Can the radius be negative?
In some mathematics contexts, negative polar radius values are allowed and can be interpreted by reversing direction through the origin. This calculator treats radius as a nonnegative distance, so \(r\) must be zero or greater.
What happens when \(r = 0\)?
When \(r = 0\), the point is at the origin. The angle does not change the location, because there is no distance away from the origin. For that reason, the displayed angle at zero radius should not be treated as a meaningful direction.
Why does the exact \(x/y\) result sometimes say there is no simple exact form?
Exact symbolic values are practical for common special angles such as \(30^\circ\), \(45^\circ\), and \(60^\circ\). For most other angles, sine and cosine do not have a simple familiar exact form, so the decimal result is usually the useful one.
Sources and References
Books
- Jay Abramson. Precalculus 2e. OpenStax, 2021. Relevant sections: 5.1 “Angles,” 5.2 “Unit Circle: Sine and Cosine Functions,” and 8.3 “Polar Coordinates.” https://openstax.org/books/precalculus-2e
- Gilbert Strang and Edwin “Jed” Herman. Calculus Volume 3. OpenStax, 2016. Relevant section: 1.3 “Polar Coordinates.” https://openstax.org/books/calculus-volume-3
Online and Official Sources
- Python Software Foundation. “math — Mathematical functions.” Python 3.14.6 documentation, accessed July 4, 2026. https://docs.python.org/3/library/math.html