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.

Why (a+b) squared= a squared + b squared + 2ab ?

Euclidian Geometry

The thing is that real math is geometry. This is one of the most ancient, usefull (the way math really is) and intuitive form of mathematics. To really figure out what`s going on you should learn geometry.
From today, 16th of october, 2011 I`ll start posting explanations for various mathematical formulas,lemmas, theorems et caetera which I know with the hope that valuable information will be preserved in a more condensed form.

Back to the problem. The area of anything is measured as its width at all points multiplyed by its length at all points (some sort of an average if you like, in mathematics that`s known as the integral, but this was invented after some time).
Therefore, the area of the square with side (let`s call it c, c being the sum of a and b) c is its width at all points which is c multiplyed by its lenght at all points which is also c. So the area of the big square is (a+b) squared (now you get it why the square power exists). Now, supose I draw paralells to the sides of the square in such a way that I divide its side in length a + length b. Now the area of the big square must equal the area of the 2 little squares with sides a and b respectively  + 2 times the area of the rectangle formed in this way.
But what is that? The area of the pieces is: a squared for the smallest square, b squared for the larger square 2 times the area of the rectangle, which is a times b (recal what area means).
All in all, by an area proof (a+b) squared must equal a squared + b squared + 2ab.