Polar and Cartesian Coordinates
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
If you have a point in Cartesian Coordinates (x,y) and need it in Polar Coordinates (r,θ), you need to solve a triangle where
you know two sides.
Example: What is (12,5) in Polar Coordinates?
Use Pythagoras Theorem to find the long side (the hypotenuse):
r^{2} = 12^{2} + 5^{2}
r = √ (12^{2} + 5^{2})
r = √ (144 + 25) = √ (169) = 13
Use the Tangent Function to find the angle:
tan( θ ) = 5 / 12
θ = atan( 5 / 12 ) = 22.6 °
So, to convert from Cartesian Coordinates (x,y) to Polar Coordinates (r,θ):
r = √ (x^{2} + y^{2})
θ = atan( y / x )
To Convert from Polar to Cartesian
If you have a point in Polar Coordinates (r, θ), and need it in Cartesian Coordinates (x,y) you need to solve a triangle where
you know the long side and the angle:
Example: What is (13, 23 °) in Cartesian Coordinates?
Use the Cosine Function for x: 
cos( 23 °) = x / 13 
Rearranging and solving: 
x = 13 × cos( 23 °) = 13 × 0.921 = 11.98 


Use the Sine Function for y: 
sin( 23 °) = y / 13 
Rearranging and solving: 
y = 13 × sin( 23 °) = 13 × 0.391 = 5.08 
So, to convert from Polar Coordinates (r,θ) to Cartesian Coordinates (x,y) :
x = r × cos( θ )
y = r × sin( θ )
And that is it !
