You've enjoyed your slice of pie, and you've even gone back for seconds while nobody was watching. You need to pack the rest into a box and take it with you to share. You can only find square-shaped boxes, how do you know you the last 2/3 of the pie is going to fit into a box?

We need to have a way to translate from polar to rectangular coordinates and vice versa. This is no problem, since we can describe where a point is using either polar or rectangular coordinates, and we only need a few tools to switch between coordinate systems.

We'll start with points in the first quadrant. Any point in the first quadrant can be described by rectangular coordinates

(x,y) where x,y≥ 0 or by polar coordinates

(r,θ) where r≥ 0 and

To translate between coordinate systems, draw a right triangle whose hypotenuse connects the origin and the point. One leg of the triangle should be on the x-axis.

The rectangular coordinates tell us the lengths of the legs of the triangle:

The polar coordinates tell us the hypotenuse and one angle of the triangle (since it's a right triangle, we know all the angles):

Each set of coordinates is telling us different information about this triangle. We can use one set of coordinates and a small selection of our tools for solving triangles
to find the other set of coordinates.

Going from rectangular to polar coordinates is the same thing as finding the magnitude and direction of a vector.

To go from polar to rectangular coordinates, remember the definitions of the sine and cosine functions (links to sine and cosine).

It's a little trickier to translate between coordinates when the points are in other quadrants.

The first thing we do, no matter which way we're translating, is recognize which quadrant the point is in. After translating between coordinates, the point should still be in the same quadrant. This will help us know if our answer is reasonable.

To go from rectangular to polar coordinates is still the same thing as finding the magnitude and direction of a vector. To find the direction we may need to use a reference angle.

To go from polar to rectangular coordinates, we don't need to worry about reference angles. All we need to do is use the definitions of sin and cos.

Translating points between rectangular and polar coordinates may be a bit tedious. Going from rectangular to polar coordinates is the same thing as
finding the magnitude and direction of a vector. Going from polar to rectangular coordinates means plugging values into the formulas

x=rcos θ
y = rsinθ

Also, if the point in question lies on the x- or y-axis, there's no need for fancy formulas. We can translate visually.

Practice: