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This topic is part of the TCS FREE high school mathematics 'How-to Library'. It shows you how to convert rectangular coordinates to polar coordinates, and vice-versa. (See the index page for a list of all available topics in the library.) To make best use of this topic, you need to download the Maths Helper Plus software. Click here for instructions.

Theory:

Rectangular coordinates and polar coordinates are two different ways of using two numbers to locate a point on a plane.

Rectangular coordinates are in the form (x,y), where 'x' and 'y' are the horizontal and vertical distances from the origin:

Polar coordinates are in the form: (r,q), where 'r' is the distance from the origin to the point, and 'q' is the angle measured from the positive 'x' axis to the point:

To convert between polar and rectangular coordinates, we make a right triangle to the point (x,y), like this:

1. Polar to Rectangular

From the diagram above, these formulas convert polar coordinates to rectangular coordinates:

x = r cosq,   y = r sinq

So the polar point: (r,q) can be converted to rectangular coordinates like this:

( r cosq, r sinq )  ð  ( x, y )

Example: A point has polar coordinates: (5, 30º). Convert to rectangular coordinates.

Solution:  (x,y) = (5cos30º, 5sin30º) = (4.3301, 2.5)

2. Rectangular to Polar

Again, from the diagram above, these formulas convert rectangular coordinates to polar coordinates:

By the rule of Pythagoras:

Tan  q = ^y/_x  ,  so therefore:

q = tan^-1( ^y/_x )

So the rectangular point: (x,y) can be converted to polar coordinates like this:

(, tan^-1( ^y/_x ) )   ð   ( r , q )

Example: A point has rectangular coordinates: (3, 4). Convert to polar coordinates.

Solution:  r = square root of(3² + 4²) = 5,   q = tan^-1(4/3) = 53.13º
                   so (r,q) = (5, 53.13º)

Method:

Maths Helper Plus can convert your coordinates from rectangular to polar or polar to rectangular, showing a helpful diagram and all of the working steps.

  

Step 2  Display the parameters box and enter values

Press the F5 key to display the parameters box:

 

Click on the edit boxes of the parameters box to enter the coordinates to convert.

 

1. To enter rectangular coordinates and convert them to polar coordinates:

 

(a) The 'X' edit box value must be '1'.

To change the 'X' value, you first need to click on the 'X' edit box.

 

(b) Click on the 'A' edit box and type the 'x' coordinate of the point.

(c) Click on the 'B' edit box and type the 'y' coordinate of the point.

(d) Click the 'Update' button to refresh the display.

2. To enter polar coordinates and convert them to polar coordinates:

 

(a) The 'X' edit box value must be '2'.

To change the 'X' value, you first need to click on the 'X' edit box.

 

(b) Click on the 'A' edit box and type the 'r' coordinate of the point.

(c) Click on the 'B' edit box and type the 'q' coordinate of the point.

(d) Click the 'Update' button to refresh the display.

Step 3  Adjust the scale of the diagram

If the graph diagram is too large to fit on the plotting area of the screen, briefly press the F10 key. This doubles the scale range in the 'x' and 'y' directions and makes the diagram smaller. Repeat if necessary.

 

To enlarge the diagram again,  hold down 'Shift' while you press F10.