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

Getting It Right

Geometry: Part 2

The Right Angle

In order to make sure your patio is squared, whether it is going to be a rectangle or a square, you will
need the cord three times the length of the distance to the distance to the first peg.

You really don't even have to measure the distance to the first peg to assure a square corner, but having a tape measure will make this easier.  Let's say your patio is to extend 12 feet (or, if you'd rather go metric, 4 meters).  This means you will need 36 feet (or twelve meters) of cord to easily square up your patio.  The cord will be a little bit longer to leave room for securing it to the pegs.

As pictured in figure #1, the tripling of the length is done by walking back and forth between the pegs with the spool of cord.

Once you have the full length (#2), walk that back to the first peg (#3) and then repeat (#4).  In the end, the four strands will be 9 feet (3 meters) long.  If you began against a wall, just lay the 9 foot piece against the wall and put the third peg down.

Fasten the 9-foot length to the third peg and the end of full cord to the first peg.  Taking the full cord, walk to the second peg, moving it to where it provides a taut line between all the pegs.

This works because in every triangle with a right angle the side opposite that angle has a length that is the square root of the sum of the squares of the other two sides.  It so happens that this ratio is found first in whole numbers with 3, 4, and 5.  And so, any multiples of these numbers produces similar triangles.  If you are going to be doing a lot of building, you could make your own "square" tool using a yardstick or even a twelve inch ruler.  Or, of course, just buy one when you get a chance!

So, what do I know?

A right triangle can be constructed based on the formula

a² + b² = c²

That is, when side c is opposite the right angle, the squares of sides a and b add up to the square of side c.

This manifests itself the ratio of 3:4:5.  That is to say, 9 + 16 = 25.  In whole numbers this only works with multiples of these three numbers.  The ratio using 1:2 would render an irrational number: the square root of 5.  Likewise 2:3 would need the square root of 13!  It only works with these three adjacent whole numbers!