Rectangular & Polar Coordinates

Rectangular says how far along and how far up; polar says how far and in what direction. Neither is more correct. Rectangular suits anything laid out on a grid, polar suits anything measured as a bearing and a distance, which is why navigation and vector work use it.

Converting to polar is Pythagoras plus an inverse tangent. The distance r is the hypotenuse of the triangle made by x and y, and the angle comes from the ratio of the two sides.

Watch the quadrant when you use tan⁻¹. A calculator returns an angle between −90° and +90°, so for points to the left of the axis you must add 180° to land in the right quadrant. This simulation does that correction, and the readout shows the true bearing.

• Try this: Set x = 3 and y = 4. What is r, and why is it a value you should recognise?
• Try this: Move the point to the second quadrant, x negative and y positive. What does θ become?
• Try this: Keep r roughly constant while changing θ. What shape does the point trace?