Areas of basic shapes, explained.

Euclidian Geometry.


The way to move on this is in the following way: Start from one shape area and then try to explain others.

The area from wich it all started is the area of the square, because its width in all places equals its lenght in all places.So we simply add up, lets say lines, of width 1 (the refferance width) and length a. So the area of the square is a times a, a squared in other words.
In an instant, on the same principle, the area of the rectanle is ab.
Now what`s the area of the parallelogram. First, construct perpendiculars from B and D to AD and BC respectively; so we have now a = b+c. Now imagine the triangle ABE shifted to the right so that AB and DC coincide. We have our area represented as a sum of the area of the BFDE rectangle and DECB rectangle. This area is dc + bd = d(c+b) = da, where d is the hight and a is the base of the rectangle.