What Is a Polar Coordinates Calculator?
A polar coordinates calculator converts points between the polar system (radius and angle) and the Cartesian system (x and y). Polar coordinates describe a point by its distance from the origin and its angle, which is convenient for circular and rotational problems. Enter a point in either system and the calculator returns the equivalent in the other.
How to Use the Calculator
- Choose the conversion direction — polar to Cartesian or Cartesian to polar.
- Enter the coordinates.
- Calculate — see the converted coordinates.
The Conversion Formulas
Polar to Cartesian:
x = r × cos(θ) and y = r × sin(θ)
Cartesian to polar:
r = √(x² + y²) and θ = arctan(y ÷ x)
where r is the radius (distance from the origin) and θ is the angle from the positive x-axis.
Worked Example
| Polar (r, θ) | Cartesian (x, y) |
|---|---|
| (5, 53.13°) | (3, 4) |
| (2, 90°) | (0, 2) |
| (1, 0°) | (1, 0) |
Polar vs Cartesian
Cartesian coordinates locate a point by horizontal and vertical distances, while polar coordinates use a distance and an angle. Polar form is natural for circles, spirals, and rotational motion, whereas Cartesian form suits straight-line and grid-based problems.
Where Polar Coordinates Are Used
- Physics: rotational and circular motion.
- Engineering: radar, antennas, and navigation.
- Mathematics: graphing circles, spirals, and complex numbers.
Frequently Asked Questions
How do you convert polar to Cartesian?
Use x = r·cos(θ) and y = r·sin(θ), where r is the radius and θ is the angle. The calculator does this for you.
How do you convert Cartesian to polar?
Use r = √(x² + y²) for the radius and θ = arctan(y/x) for the angle, taking care to place the angle in the correct quadrant.
What are polar coordinates?
Polar coordinates describe a point by its distance from the origin (r) and its angle from the positive x-axis (θ), instead of x and y.
When are polar coordinates useful?
They are convenient for circular, spiral, and rotational problems, and for graphing certain curves that are awkward in Cartesian form.
Is this polar coordinates calculator free?
Yes — it is completely free, requires no signup, and converts both directions.