A Square Deal

Geometry 4:

A Square Deal

When working with the right triangle, it is easy to build a "rectangle" by simply building a mirror image of the right triangle.  One might expect the term "right rectangle" to be used here, but that would be redundant.  This is because the prefix "rect" is a corruption of the original Germanic form of the Latin "rectus" which means "right."  The Old German was "reht" but the "h" was the hard "kh" sound that migrated up from the Greek "Chi" (looks like our "X").  The Old English form was riht, corrected (pun intended) by adding back in the hard sound.  This time, the hard "g" was used, rendering "ri-ght"  It did not take long before the hardness of the "chi" sound was abandoned.  From "rect" to "right" and back again!

Anyway, we have seen how a right angle can be constructed using the 3:4:5 ratio with the sides.  Using the largest side (the diagonal "5") a ratio of 5:4:3 becomes the mirror image forming a 3 by 4 rectangle.  The area of the rectangle is simple to determine in "square" units.  The area of a "right quadrangle" (four-sided with parallel sides) will be twice that of a right triangle.  Another way of seeing this is that the formula for triangle will be one half that of a rectangle.

The formulae for this are simple:

Area of a rectangle = base (width) x height (length).
Area of a triangle = 1/2 x base x height

The "square" is a special kind of rectangle.  Not only are all the angles inside a square equal, but so are the sides.  The ratio of the sides of the "half square" loses its whole number diagonal when "side a" and "side b" are forced to be the same size.  The diagonal becomes a multiple of the square root of 2! That works out to a little over 1.414.  I'd go out farther, but fourteen-fourteen works for me. But, when considering the area of the square, no irrational numbers need be involved.  I really prefer rational numbers.

When we move to other triangles and polygons, where the angles are not all right (no pun intended), then the "height" becomes a problem.  The height of a triangle is always figured in a right angle, which must be used to divide the shape in order to follow the rules of the square.  That is to say, everything must be reduced to right triangles and/or rectangles to be squared.  It's like playing with blocks!

I'll work out the bugs of those building blocks in another blog.  I know just enough geometry to be "dangerous."

So, what do I know?

Rectangle: Area = lw
Rt. Triangle:  Area = (1/2)lw = lw/2, where l and w form the right angle.

Square: area = l² (length squared) [length being equal to width)

All area within polygons are measured by the pythagorean theorem:  a² + b² = c²