Polar to Cartesian Calculator (Polar Coordinates)

This free polar to Cartesian calculator converts polar coordinates (a radius and an angle) into Cartesian X and Y coordinates, and the other way around. It can also work from an origin other than 0,0, such as the center of a bolt circle:

For example, suppose we have Cartesian Coordinates of X = 10 and Y = 15. Switch the calculator to Cartesian to Polar and enter them, and you get a radius of 18.0278 at an angle of 56.3099 degrees. Switch back to Polar to Cartesian with those values and you're back at X = 10, Y = 15.

What are Polar Coordinates vs Rectangular Coordinates (Cartesian Coordinates)?

We're all used to Cartesian coordinates (rectangular) involving X, Y, and Z.  Polar coordinates consist of only 2 values:

• An angle or bearing to the desired point taken from some origin point.
• A distance or radius to be followed along the bearing angle to go from the origin point to the destination point.

A Polar Coordinate System is useful for solving problems that involve subdividing a region into segments of one kind or another.  For example, when figuring out a bolt circle.

Polar Coordinates are good for solving problems in 2D space.  For three dimensions applications you may need Cylindrical Coordinates.

Typically, we will operate most of the time with Cartesian Coordinates since they are more familiar and only switch to Polar Coordinates for special cases and problems.  Having both polar and cartesian coordinates available is really handy, so it is good to be able to easily convert between the two.

Math behind Converting Polar to Cartesian Coordinates

Converting from Polar to Cartesian coordinates is easy to do. Refer to this diagram:

Variables used in polar form coordinates and conversions...

The variables used for polar and rectangular coordinates are:

Cartesian Coordinates:

x: The X-Axis coordinate

y: The Y-Axis coordinate

Polar Coordinate System:

r: The radius or distance on the polar vector

Theta θ (Greek character): The polar angle from the X-Axis in a counter-clockwise direction.  This angle is measured about the polar axis which is at (0, 0) in rectangular coordinates.

Equation to Convert Polar to Cartesian Coordinates

Given r and Theta, we can find x and y to convert from Polar to Rectangular coordinates as follows:

Converting Polar to Cartesian Coordinates

x = r Cos( Theta )

y = r Sin( Theta )

Equation to Convert Rectangular to Polar Coordinates

If we know x and y, we can find r and Theta to convert from Cartesian Coordinates to Polar coordinates as follows:

Converting Cartesian Coordinates to Polar Coordinates (Rectangular to Polar Calculator)

r = SQRT( x * x + y * y )

Theta = arctan( y / x )

One catch: arctan( y / x ) only gives the right angle when x is positive. For points to the left of the Y axis, add 180 degrees (and add 360 to any negative result if you want angles from 0 to 360). Spreadsheets and programming languages have an ATAN2(y, x) function that handles the quadrants for you, and it's what the calculator above uses.

Video on How to Convert With a Hand-Held Calculator

Polar Coordinates Application to CNC and Bolt Circles

Programming a CNC machine in polar coordinates can simplify a lot of problems.  Take bolt circles for example.  They're very easy to do in Polar Coordinates.  The radius of the circle is the polar radius.  And the angle between holes is just 360 degrees divided by the number of holes.

Whether you want to program the bolt circle directly in G-code polar coordinates or use a calculator like this one to crank out the normal Cartesian coordinates, bolt circles are a lot easier when done this way.

For more on polar coordinates G-code programming see our article:

• G15 and G16 G-codes [Polar Coordinates and CNC Bolt Circles ]

Conclusion

Being able to use both polar and Cartesian coordinates can be a helpful time saver.  Polar coordinates are a natural way to set up many problems such as bolt circles.  Cartesian coordinates are more familiar, and will still be more commonly used than polar coordinates.  In this article we've given you the tools to convert polar coordinates to the cartesian coordinate system and vice versa.

Other Resources

• More detail, examples, and a quiz on Rectangular to Polar Conversions

Can all Cartesian Coordinates be written as Polar Coordinates

Yes, every point in the Cartesian plane (x, y) has corresponding Polar Coordiantes.

What is the polar form of the Cartesian point (0, 0)?

The polar form for Cartesian (0, 0) is not unique because every angle of theta coupled with 0 radius can represent that Cartesian coordinate.  However, the convention is to use (0, 0) for the polar form.

How do you find the polar coordinates?

If we know x and y, we can find r and Theta to convert from Rectangular to Polar Coordinates as follows: r = SQRT( x * x + y * y ) Theta = arctan( y / x ) `