In geometry class in high school, I was always given the area formulas for 2d and 3d shapes. But I had no explanation for how these formulas came to be. Understanding these formulas was never expected of me, but it was something that I expected to be taught. If you only came to this blog to look at the area formulas, I encourage you to read on and learn more math. Here is an image that showcases many different area formulas.
Let’s understand these formulas now.
How do we bring shapes from a simple form into a more complicated form: a 1d line to a 2d shape, or a 2d shape to a 3d solid? We multiply. By something. It’s as simle as that in ALL cases. The simplest case is a square.
We start with a line, and if we want to extend the line into a new dimension, we multiply by how much we want to extend it by, creating a rectangle (in this case a square). Then to turn a square into a cube, we multiply by how much we want to extend the square by. Notice that this is the same explanation for if you wanted to make a rectangle and a recangular prism.
You ponder, “Surely, it can’t be that simple!”
“Oh, but it can! And don’t call me Sherley.”
“Fine. Maybe for a square this works (that’s the easy one) but… what about circles?”
We start with a line, and if we want to extend the line into a new CIRCULAR dimension, we multiply by how much we want to extend it by. Circles, however use a special multiplier (you might have heard of pi). So to create a circle from a line (assuming the line is our radius), we multiply the length of the line times itself times π. To turn a circle into a cylinder, we multiply by how tall we want to extend the circle by just like we did with a square to a cube or a rectangle to a rectangular prism.
To make a circle a sphere… we multiply the area of or circle by (4/3) times the radius. But why 4/3? And what even is pi?!?! Find out next time!