Programmers and Teachers:
Rectangular to Polar Calculator
Rectangular and Polar Coordinates Lesson
Rectangular Coordinates vs. Polar Coordinates
Notated as (x, y), rectangular coordinates (also called Cartesian coordinates) give us the position of a point in terms of its location relative to the x-axis and y-axis. Notated as (r, θ), polar coordinates give us the position of a point with radius measured as a distance away from the origin and angle measured counter-clockwise off the positive x-axis (also called the polar-axis or r-axis).
Imagine the polar coordinate angle θ is measured on a clock. 0° is 3 o’clock, 90° is 12 o’clock, 180° is 9 o’clock, and 360° is back to 3 o’clock. If the angle is negative, go clockwise. For example, -90° is at the 6 o’clock position.
Both styles of coordinates have their strengths and weaknesses and are best used in different situations. Rectangular coordinates are very easy to use for polynomial functions that use x and y as variables. For functions that use r and θ as variables, polar coordinates are easier to work with. The formulas for converting rectangular to polar are given as:
Pay very close attention to the units used when evaluating tan-1(y/x) to find θ. If using a calculator, make sure the output unit on the calculator matches the unit written next to the answer. If we calculate in radians but write degrees as the unit next to our angle it is incorrect.
Convert the rectangular coordinates (3, 6) to polar coordinates. Express the polar angle in degrees.
1.) First, let’s find our radius coordinate. Applying the formula gives us:
r = (32 + 62) = 6.708
2.) Now let’s find the angle coordinate. Applying the formula gives us:
θ = tan-1(6/3) = 63.435°
3.) Our point in polar coordinates is (r, θ) = (6.708, 63.435°).
How the Rectangular to Polar Calculator Works
The calculator at the top of this page uses the exact same two equations as listed above. The input coordinates are sent to a computer and run through a program that applies the x and y values to those two equations.
The polar radius and angle are then sent back here and displayed. This all happens very quickly because a computer processor is doing the work. However, as humans, we can easily hand calculate the conversion with paper and pencil algebra and a scientific calculator just as accurately!