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Polar Coordinate System

When each point on a plane of a two-dimensional coordinate system is decided by a distance from a reference point and an angle is taken from a reference direction, it is known as the polar coordinate system.

Pole = The reference point

Polar axis = the line segment ray from the pole in the reference direction

In the polar coordinate system, the origin is called a pole.

Polar coordinates

Here, instead of representing the point as (x, y), we can express it as a polar coordinate (r, θ).

Where the value of r can be negative. The value of angle changes based on the quadrant in which the r lies.


Value of


Calculated value


Add π to the calculated value


Add π to the calculated value


Add π to the calculated value

Read more:

The above information can be tabulated as below:


Cartesian Coordinates


Quadrants in the Cartesian plane


(x, y)



(-x, y)



(-x, -y)



(x, -y)



In the Cartesian coordinate system, the distance of a point from the y-axis is called its x-coordinate and the distance of a point from the x-axis is called its y-coordinate.

Polar grid

Polar grid with different angles as shown below:

Polar grid

Also, π radians are equal to 360°.

Polar Coordinates Formula

We can write an infinite number of polar coordinates for one coordinate point, using the formula

(r, θ+2πn) or (-r, θ+(2n+1)π), where n is an integer.

The value of θ is positive if measured counterclockwise.

The value of θ is negative if measured clockwise.

The value of r is positive if laid off at the terminal side of θ.

The value of r is negative if laid off at the prolongation through the origin from the terminal side of θ.


The side where the angle starts is called the initial side and the ray where the measurement of the angle stops is called the terminal side.

Cartesian to Polar Coordinates

x = r cos θ

y = r sin θ

cartesian to polar coordinate

Finding r and θ using x and y:

Finding r and θ using x and y

3D Polar Coordinates

3d polar coordinates or spherical coordinates will have three parameters: distance from the origin and two angles.

The 3d-polar coordinate can be written as (r, Φ, θ).


R = distance of from the origin

Φ = the reference angle from XY-plane (in a counter-clockwise direction from the x-axis)

θ = the reference angle from z-axis

3d polar coordinates

Polar Coordinates Examples

Example 1:

Convert the polar coordinate (4, π/2) to a rectangular point.



Polar coordinate equation

We know that,

Polar coordinate Example 1

Polar coordinate Example 2

Hence, the rectangular coordinate of the point is (0, 4).

Example 2:

Convert the rectangular or cartesian coordinates (2, 2) to polar coordinates.



(x, y)=(2, 2)

Polar coordinate Examples 3

Polar coordinates examp 5


Polar coordinate Notes 1

Polar Coordinates Applications

Two-dimensional polar coordinates play a significant role in navigation either on sea or in air.

If equations are expressed in polar coordinates, then calculus can be applied.

Test your Knowledge on Polar Coordinates

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