How to convert rectangular coordinates to polar coordinates, and viceversa.
Goal:
Theory:
Part 1
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, Θ), where ‘r’ is the distance from the origin to the point, and ‘Θ’ 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, Θ) 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 Θ = 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, Θ = tan1(4/3) = 53.13º
so (r,Θ) = (5, 53.13º)
Method:
Part 2
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 1 Download the free support file…We have created a Maths Helper Plus document containing the completed example from this topic. You can use this to practice the steps described below, and as a starting point for solving your own problems.
File name: ‘Coordinate converter.mhp’ File size: 21kb
Click here to download the file.If you choose` ‘Open this file from its current location’, then Maths Helper Plus should open the document immediately. If not, try the other option: ‘Save this file to disk’, then run Maths Helper Plus and choose the ‘Open’ command from the ‘File’ menu. Locate the saved file and open it. If you do not yet have Maths Helper Plus installed on your computer, click here for instructions.
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.

 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.

 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 ‘Θ’ coordinate of the point.
(d) Click the ‘Update’ button to refresh the display.
Step 3 Adjust the scale of the graph
If the graph scale is too small to display all of the points, briefly press the F10 key. This doubles the scale range in the ‘x’ and ‘y’ directions. Repeat until all plotted points are visible.
To reduce the scale again, hold down ‘Shift’ while you press F10.
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