Polar And Cartesian Coordinates - Math Is Fun

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):

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

Use the Tangent Function to find the angle:

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

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 a ratio (like 512)Inverse Tangent takes a ratio and gives an angle (like 22.6°)

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 × 0.923x = 12.002... Use Sine for y:sin( 22.6° ) = y / 13 Rearrange and solve:y = 13 × sin( 22.6° )y = 13 × 0.391y = 4.996...

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

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

• x = r × cos( θ )
• y = r × sin( θ )

How to Remember Which is Which?

(x,y) is alphabetical, (cos,sin) is also alphabetical

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

And in JavaScript (if you like coding):

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:

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

r = 12 and θ = 195°

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

(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

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 2167, 2168, 2169, 2170, 2171, 2172, 2173, 2174, 5159, 5160 Interactive Cartesian Coordinates Cartesian Coordinates Graphs Index Geometry Index Copyright © 2025 Rod Pierce