Polar and Cartesian Coordinates

... and how to convert between them.

To pinpoint where you are on a map or graph there are two main systems:

Cartesian Coordinates

Using Cartesian Coordinates you mark a point by how far along and how far up it is:

Polar Coordinates

Using Polar Coordinates you mark a point by how far away, and what angle it is:

Converting

To convert from one to the other, you need to solve the triangle:


To Convert from Cartesian to Polar

When you know a point in Cartesian Coordinates (x,y) and want it in Polar Coordinates (r,θ) you solve a right triangle with two known sides.

Example: What is (12,5) in Polar Coordinates?

Use Pythagoras Theorem to find the long side (the hypotenuse):

r2 = 122 + 52
r = √ (122 + 52)
r = √ (144 + 25) = √ (169) = 13

Use the Tangent Function to find the angle:

tan( θ ) = 5 / 12
θ = tan-1 ( 5 / 12 ) = 22.6° (to one decimal)
calculator-sin-cos-tan

What is tan-1 ? It is the Inverse Tangent Function.

  • Tangent takes an angle and gives you a ratio,
  • Inverse Tangent takes a ratio (like "5/12") and gives you an angle.

Answer: the point (12,5) is (13, 22.6°) in Polar Coordinates.

 

So to convert from Cartesian Coordinates (x,y) to Polar Coordinates (r,θ):

  • r = √ ( x2 + y2 )
  • θ = tan-1 ( y / x )

Note: Calculators may give the wrong value of tan-1 () when x or y are negative ... see below for more.

To Convert from Polar to Cartesian

When you know a point in Polar Coordinates (r, θ), and want it in Cartesian Coordinates (x,y) you solve a right triangle with a known long side and angle:

Example: What is (13, 22.6°) in Cartesian Coordinates?



Use the Cosine Function for x:   cos( 22.6 °) = x / 13
Rearranging and solving:   x = 13 × cos( 22.6 °) = 13 × 0.923 = 12.002...
     
Use the Sine Function for y:   sin( 22.6 °) = y / 13
Rearranging and solving:   y = 13 × sin( 22.6 °) = 13 × 0.391 = 4.996...

Answer: the point (13, 22.6°) is almost exactly (12, 5) in Cartesian Coordinates.

 

So, to convert from Polar Coordinates (r,θ) to Cartesian Coordinates (x,y) :

  • x = r × cos( θ )
  • y = r × sin( θ )

 

But What About Negative Values of X and Y?

Four Quadrants

When we include negative values, the x and y axes divide the
space up into 4 pieces:

Quadrants I, II, III and IV

(They are numbered in a counter-clockwise direction)

Quadrants

When converting from Polar to Cartesian coordinates it all works out nicely:

Example: What is (12, 195°) in Cartesian coordinates?

r = 12 and θ = 195°

  • x = 12 × cos(195°) = 12 × -0.9659... = -11.59 to 2 decimal places
  • y = 12 × sin(195°) = 12 × -0.2588... = -3.11 to 2 decimal places

So the point is at (-11.59, -3.11), which is in Quadrant III

But when converting from Cartesian to Polar coordinates,

... the calculator may give you the wrong value of tan-1

It all depends what Quadrant the point is in! Use this to fix things:

Quadrant Value of tan-1
I Use the calculator value
II Add 180° to the calculator value
III Add 180° to the calculator value
IV Add 360° to the calculator value

Example: P = (-3, 10)

P is in Quadrant II

  • r = √((-3)2 + 102) = √109 = 10.4 to 1 decimal place
  • θ = tan-1(10/-3) = tan-1(-3.33...)

The calculator value for tan-1(-3.33...) is -73.3°

The rule for Quadrant II is: Add 180° to the calculator value
θ = -73.3° + 180° = 106.7°

So the Polar Coordinates for the point (-3, 10) are (10.4, 106.7°)

Example: Q = (5, -8)

Q is in Quadrant IV

  • r = √(52 + (-8)2) = √89 = 9.4 to 1 decimal place
  • θ = tan-1(-8/5) = tan-1(-1.6)

The calculator value for tan-1(-1.6) is -58.0°

The rule for Quadrant IV is: Add 360° to the calculator value
θ = -58.0° + 360° = 302.0°

So the Polar Coordinates for the point (5, -8) are (9.4, 302.0°)

 

Summary

To convert from Polar Coordinates (r,θ) to Cartesian Coordinates (x,y) :

  • x = r × cos( θ )
  • y = r × sin( θ )

To convert from Cartesian Coordinates (x,y) to Polar Coordinates (r,θ):

  • r = √ ( x2 + y2 )
  • θ = tan-1 ( y / x )

The value of tan-1( y/x ) may need to be adjusted:

  • Quadrant I: Use the calculator value
  • Quadrant II: Add 180°
  • Quadrant III: Add 180°
  • Quadrant IV: Add 360°