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Polar Coordinates Calculator

Make conversions between rectangular (cartesian) and polar plane coordinates.

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Polar Coordinates Calculator

This calculator is used to convert rectangular to polar coordinates and vice versa for a 2D system. It shows the step-by-step calculation for both conversions.

What are Cartesian and Polar Coordinates?

cartesian

1.1 Polar Coordinates:

Polar coordinates represent a point that is positioned with respect to the origin (0,0) and an angle from the origin in the reference direction. They are written as (r, θ), such that:

  • r is the radial distance from the origin to point
  • θ is the angle between the x-axis and the line joining point with the origin.

1.2 Cartesian (Rectangular) Coordinates:

Cartesian coordinates indicate a point in an XY-plane by a pair of numerical values. These coordinates are signed distances from a point to two perpendicular directed lines. They are usually written as (x, y), such that:

  • X represents the horizontal distance from the origin (positive to the right, negative to the left).
  • Y represents the vertical distance from the origin (positive upwards, negative downwards

How to Make Conversion Between Cartesian and Polar Coordinates?

There are two methods of converting between the coordinates:

2.1 Manual Calculations:

2.1.1 To Convert from Cartesian to Polar

cartesian to polar coordinate conversion

For rectangular coordinates (x, y) given in the above picture, you can find polar coordinates (r, θ) as follows:

\(r = \sqrt{x^{2} + y^{2}}\)

\(θ = arctan (\dfrac{y}{x}\)

2.1.2 To Convert from Polar to Cartesian

polar to cartesian coordinate conversion

Use the following polar equations to cartesian equations for converting: \(x = r cos θ\) \(y = r sin θ\)

2.2 Using Polar Coordinates Calculator

  1. Select the conversion type
  2. Enter the required coordinate values
  3. Click ‘Calculate’ and get converted form

Cartesian & Polar Coordinates (Solved Examples)

Example 01:

cartesian to polar example

Solution:

Data Given:

  • Vertical coordinate = y = 2
  • Horizontal coordinate = x = 6

Calculations:

Step 1: Determine ‘r’

\(r = \sqrt{x^{2} + y^{2}}\)

\(r = \sqrt{6^{2} + 2^{2}}\)

\(r = \sqrt{36 + 4}\)

\(r = \sqrt{40}\)

\(r = 6.324555320336759\)

Step 1: Determine ‘θ’

\(θ = arctan (\dfrac{y}{x}\)

\(θ = arctan (\dfrac{2}{6}\)

\(θ = arctan (\dfrac{1}{3}\)

Example 2

cartesian to polar example

Solution:

Data Given:

  • r = 2
  • \(θ = 45^\text{o}\)

Calculations:

Step 1: Determine Horizontal Rectangular Component ‘x’

\(x = r cos θ\)

\(x = 2 * 0.707\)

\(x = 1.414\)

Step 2: Determine Vertical Rectangular Component ‘y’

\(y = r sin θ\)

\(y = 2 * sin\left(45^\text{o}\right)\)

People Also Ask:

Are polar coordinates unique?

No, the polar coordinates of a point are not unique. The reason is that each point can be represented by infinite polar coordinates in infinite ways.

What is z in polar coordinates?

In polar coordinates, z represents the complex number in polar form, such that: z = x + iy (where i (iota) = \(\sqrt{-1}\)

Where

  • x = Real part
  • y = Imaginary part

What is 1 in polar coordinates?

The polar coordinate representation of 1 is given as:

\(1 = 1e^{i} 0\)

The same equation holds true for \(1 = 1e^{i} 2π\) due to the fact that sine and cosine are both periodic with the angle 2π.

Sample Conversions Between Cartesian and Polar Coordinates

Cartesian Coordinates (x, y) Polar Coordinates (r, θ)
(3, 4) √(25) , arctan(4/3) ≈ 53.13°
(-2, 1) √(5) , arctan(-1/2) ≈ -26.57°
(0, 5) 5, 90°
(-4, 4) √(32) , arctan(1/1) ≈ 45° (or 315°)

This polar coordinates calculator can handle all the conversions between coordinates including all ones listed in the above table, with the complete solution shown.

Reference: 

From the source of Wikipedia: Polar coordinate system, Conventions, Uniqueness of polar coordinates, Converting between polar and Cartesian coordinates.

From the source of Math Insight: Conversion formulas, the plane of Cartesian coordinates, coordinates r and θ.

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