Why does the square and the circle's angles add up to 360 degrees?

Answer 1

The short answer is "They Don't!"

First we must understand what a convention is. A convention is simply an agreement that we all follow. Sometimes there are many conventions. E.g. We measure speed in kph, mph, mach and knots to name a few. In some cases the whole world agrees on a single convention.

By convention a circle is divided into 360 slices, each slice is called "one degree". Thus a circle is divided into 360 degrees and now we have a tool to measure angles. This convention is world wide.

A square is a square because it has 4 equal sides and 4 equal corners. To measure a square or any polygon, we move along the boundary and using our measuring device we measure every change of direction. A square measures 360 degrees.

If we measure a circle using the same rules that we apply to polygons we must start at a point on the circumference and measure each change of direction. A circle is always changing, thus we can say that a circle has an infinite number of changes of direction. Each change is infinitely close to 180 degrees. Thus a circle has an infinite number of degrees when measured using the same rules that we apply to polygons.

Consider the following test given by teachers to children.

How many degrees in a triangle___________________Answer___180

How many degrees in a square____________________Answer___360

How many degrees in a hexagon__________________Answer___720

How many degrees in a octagon__________________Answer___1080

How many degrees in a circle____________________Answer___Infinity

This test sets up a mathematical progression for young minds to solve. The answer to the last question is not 360. It is in fact infinity.

If a teacher wants an answer of 360 then the question should be a separate question such as "how many degrees is a circle divided into?