Polar and Cartesian Coordinates

... and how to convert between them.

In a hurry? Read the Summary. But please read why first:

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

Cartesian Coordinates

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

Polar Coordinates

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

Converting

To convert from one to the other we will use this triangle:

To Convert from Cartesian to Polar

When we know a point in Cartesian Coordinates (x,y) and we want it in Polar Coordinates (r,θ) we 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):

Use the Tangent Function to find the angle:

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

What is tan^-1 ?

It is the Inverse Tangent Function:

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

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

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

Use Cosine for x:cos( 22.6° ) = x / 13

Rearrange and solve:x = 13 × cos( 22.6° )

x = 13 × 0.923

x = 12.002...

Use Sine for y:sin( 22.6° ) = y / 13

Rearrange and solve:y = 13 × sin( 22.6° )

y = 13 × 0.391

y = 4.996...

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

How to Remember Which is Which?

Also "y and sine rhyme" (try saying it!)

		function toCartesian(r, angle) {
			let x = r * Math.cos(angle)
			let y = r * Math.sin(angle)
			return [x, y]
		}	

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:

But when converting from Cartesian to Polar coordinates ...

... the calculator can give the wrong value of tan^-1

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

(Yes, both Quadrant's II and III are "Add 180°")

Example: P = (−3, 10)

P is in Quadrant II

• r = √((−3)² + 10²)
  r = √109 = 10.4 to 1 decimal place
• θ = tan^-1(10/−3)
  θ = 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)
  r= √89= 9.4 to 1 decimal place
• θ = tan^-1(−8/5)
  θ = 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