Perimeter and City Blocks

At some point in your life you were probably asked a question like this: what is the perimeter of this shape?

This is known as a "composite" shape, and kids are taught to find the perimeter of such a thing by looking at it as a bunch of rectangles stuck together, finding the length of each little line, and then adding the lengths. This is a fine method, works great. However, when the shapes are particularly nice, there's a much easier way.

This easier way is based on how we measure distances in a city. Assuming we can't cut through buildings, we measure distances in a city by blocks. In the picture below, the distance from A to B is 7 blocks. Any reasonable path from A to B must go a total of 4 blocks East and 3 blocks North. We don't have to do all the traveling East at once - we might go 1 block E, then 3 blocks N, then 3 blocks E. We've still travelled a total of 4 blocks E and 3 blocks N.

This brings us back to the original question, how to find the perimeter of the composite shape. We can think of this shape as being made of two paths from A to B, where each path uses only the directions E and N.

Looking at the "lower" path, we see that to travel from A to B one must go a total of 24 blocks E and 15 blocks N. Although we aren't given any distances on the "upper" path, we know that the red line segments on the upper path must take us the same distance East as the red line segments on the lower path (24 blocks). Similarly, the black line segments on the upper path must take us the same distance North as the black line segments on the lower path (15 blocks).

The upper and lower path each go 24 blocks E and 15 blocks N, so the total perimeter is

2(24+15)=2(39)=78.

Warning: this trick is just for shapes made up of two paths that only travel East and North!