Kalid,

You might want to make it clear that you’re restricting your discussion to squares, since the statement

The area of any shape can be computed from any line segment squared.

is not true for rectangles.

Kalid,

You might want to make it clear that you’re restricting your discussion to squares, since the statement

The area of any shape can be computed from any line segment squared.

is not true for rectangles.

Actually, I think I see what you mean. (Sorry, I’m commenting as I read. Maybe I should write down my comments and not say them until I’m finished reading, huh?)

I would like to see a derivation of mccoyn’s result of c = a * sqrt(1 + b^2 / a^2) being equivalent to c = a^2 + b^2, if anyone has one.

Hi John, thanks for the comments! No worries about the inline comments, it’s interesting seeing the thought process. Yep, rectangles can still follow the rule constant * side squared, but that constant will be different for each shape. In the case of a square, the constant is 1 (it is a different way to look at it).

The second result

c^2 = a^2 + b^2

c^2 = a^2 * (1 + b^2/a^2) [rearrange right side]

c = a * sqrt(1 + b^2/a^2) [square root of both sides]

Now, the physical meaning of this is interesting. It basically gives you a constant [sqrt(1 + b^2/a^2)] that maps you distance in the “a” direction to relative distance in the “c” direction.

There’s more details here if you like:

http://betterexplained.com/articles/rescaling-the-pythagorean-theorem/

In the section on proving the Pythagorean Theorem (Intuitive Look at the Pythagorean Theorem), I am unsure why the Area = F*hypotenuse^2. Can you please explain? I tried to use the A= 1/2bh equation and substitute one of the sides with h = sqrt(hyp^2 - B^2) but could not come back to the F * hyp^2.

Thanks,

Ashley

@Ashley: Hi, that equation

Area = F*hypotenuse^2

refers to the fact that any triangle can have an equation formula like this. The amount of F will change on the shape though.

Area = 1/2 b * h

is a more useful equation because it works for every type of triangle. But the first one gets at the idea that the area of any shape is essentially based on the hypotenuse squared (or any side squared, for a different F). For squares, area is 1 * side^2, or 1/16 * perimeter^2. In both cases, it’s “some number time a measurement squared”.

Hope this helps!

Could u please explain me the concept of Area factor. Im really not getting it. With respect to various shapes how can i associate it with the given figure.