Enter your rectangular coordinates or polar coordinates into the calculator to convert between the two forms.

Conversion Formula

The following formulas are used to convert between rectangular and polar coordinates. 

r = √(x² + y²)
θ = atan2(y, x)
x = r * cos(θ)
y = r * sin(θ)

Variables:

  • r is the magnitude
  • θ is the angle in degrees
  • x and y are the rectangular coordinates

To convert rectangular coordinates to polar form, use the formulas for r and θ. To convert polar coordinates to rectangular form, use the formulas for x and y.

What is Polar Form?

Polar form is a way of expressing complex numbers or coordinates in terms of their magnitude and angle. This form is particularly useful in fields such as physics, engineering, and navigation, where it’s often easier to work with magnitudes and directions rather than Cartesian coordinates.

How to Convert Between Forms?

To convert between rectangular and polar coordinates, follow these steps:

  1. For rectangular to polar, calculate the magnitude using the square root of the sum of the squares of the x and y coordinates.
  2. Calculate the angle using the arctangent of y divided by x.
  3. For polar to rectangular, calculate the x coordinate using the magnitude multiplied by the cosine of the angle.
  4. Calculate the y coordinate using the magnitude multiplied by the sine of the angle.
  5. Check your answers using the calculator above.

Example Problem:

Use the following variables as an example problem to test your knowledge.

Rectangular Coordinates: x = 3, y = 4

Polar Coordinates: r = 5, θ = 53.13°

FAQ

1. What is the magnitude in polar coordinates?

The magnitude is the distance from the origin to the point, calculated as the square root of the sum of the squares of the x and y coordinates.

2. How is the angle calculated?

The angle is calculated using the arctangent function, which gives the angle whose tangent is the ratio of y to x.

3. Why use polar coordinates?

Polar coordinates are useful in situations where direction and distance are more important than horizontal and vertical positions, such as in navigation and engineering.

4. Can this calculator handle negative coordinates?

Yes, the calculator can handle negative coordinates and will correctly compute the magnitude and angle.

5. Is the calculator accurate?

The calculator provides an estimate based on the inputs provided. For exact figures, particularly in critical applications, it’s best to verify with additional tools or methods.