Emerald City Comic Con (ECCC) is this weekend, and, while I can’t attend myself (I work weekends and I always wait till GeekGirlCon to request time off), I’ve been helping a friend with her Lady Sif costume. With no pattern to work from, our primary job was to look at photos and make posterboard templates of Lady Sif’s armor, which were then traced onto thin foam, cut out, and covered in metallic fabric.
She’s an artist and I’m definitely not. But my experience as a math tutor means I can draw polygons decently, so I started with the six quadrilateral abdominal plates. When we laid the first few templates that I made against the base of the costume, we found that I had the shape right. However, the size was definitely off. I needed to make the templates smaller, but I didn’t want to change the overall shape- I wanted my new shape to be similar to my old one. When two polygons are similar, all of the angles in one shape are the same as the angles in the other shape, and the length of each side has been multiplied (or divided) by the same factor.
I trimmed the appropriate amount off of each side. While the result looked good against the base of the costume, my friend noted that the new template was a lot smaller than the old one. Intuitively, we think that whatever happens to the side lengths of a polygon should be the same thing that happens to the area of the polygon- if each side length is cut in half, the area should be cut in half, right?
Let’s pretend we had a rectangle instead of a parallelogram. The area of a rectangle is length * width.
Say we doubled the length of each side to 2a and 2b.
The area of this rectangle is four times the area of our original rectangle.
We can do the same visual comparison with the quadrilateral we started with; since each side length was divided by 2, the entire area was divided by 4.
Geometry is full of surprises.
Another geometric surprise came up while we were measuring armbands for the costume: the distance around a roughly circular object (its circumference) is probably greater than you think it is. The circumference of a circle is pi multiplied by its diameter- so a little more than 3 times the distance across that circle. Take a couple of drinking glasses out of your cupboard. Which would you guess is greater, the circumference of each glass or the height of each glass? Take a tape measure/length of ribbon or string and see for yourself.