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)

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)

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°

Activity: A Walk in the Desert 2