# Polar Coordinates Calculator

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

## 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?

### 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

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

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:

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

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’

$$y = r sin θ$$

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

### 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.