Every kid knows "Pythagoras' Theorem"...most can parrot it decades later "The square on the hypotenuse equals the sum of the squares on the other two sides".

What the heck does it mean?

It means that if you have a triangle with a right angle in one corner - then if you know the lengths of two sides of the triangle, you can always find the length of the third side.

This was invented by a Greek guy named "Pythagoras" about 2,500 years ago. But it's amazing how many people have it memorised - but have no clue what it's for.

This is a true story:

We all know that advancements in technology can cost people their jobs. However, in the case of the building industry in Texas, the effect of introducing new technology can often be somewhat delayed.

Back in 1997, my new house was in the slow process of changing from plans on paper into bricks on concrete. One of the tasks that has to be done early on is to lay out the shape of the house accurately onto the land. My builder uses a sub-contractor to do that - and I had occasion to watch him work. He arrived in a beat up old pickup truck with four 'migrant workers' sitting in the back. In order to lay out the initial 'bounding rectangle' of the building, they follow this algorithm:

- Measure a baseline for the long edge of the rectangle. Mark it with two stakes hammered into the ground and tie a length of nylon string between them.
- Tie a second piece of string to one of the stakes and measure out the width of the rectangle along it. Eyeball the angle between the new edge and the baseline so it's roughly 90 degrees and you have an 'L' shape. One guy holds the string there.
- Do the same at the other end of the baseline. Now you have a 'U' shape and two guys are holding the open ends of the strings.
- Take a third piece of string - equal in length to the length of the rectangle. Give one end to each of the two guys who are already holding string. 'jiggle' them until all three strings are tight. You now have a parallelogram made of string, staked out at two corners.
- Now take
**two**long tape measures and with one guy standing at each corner of our parallelogram, position the tape measures along the two diagonals of the parallelogram. With two guys holding the tapes on the baseline stakes and the other two holding onto the strings and shouting out the lengths of the diagonals, they jiggle the two free points until all of the strings are tight and the two diagonals tape measures are reading the same lengths. This requires a lot of shouting, cursing and everyone telling everyone else which way to move. - Now they have a rectangle - so they bash in two more stakes and then level the whole thing with a really impressive-looking laser contraption.

Well, I watched this with some amusement - and asked why they didn't just calculate the length of the diagonal. The boss guy said that you couldn't do that - "It's impossible". I told him about Pythagoras' theorem. With the aid of a calculator (he didn't know what that funny 'square-root' key was for), I was able to show him how easy it is to **calculate** the length of the diagonal and do away with all the ugly 'jiggling'.

- Take the length of the rectangle - multiply it by itself - write it down on a piece of scrap 2x4 (hey, the calculator has a memory button...but perhaps that's a step too far!)
- Take the width of the rectangle - multiply it by itself.
- Add the two together.
- Push the mysterious "square root" button - and that's the length of the diagonal.
- We tried it on my house - and the answer was within an inch of what the "jiggling with tapes and yelling" approach achieved.

"Wow!" he said. Then he thought for a moment - "Now I'll only need three guys to hold the string!"...and fired one of them on the spot! I thought he was kidding - but the next day when they were measuring out the place for the garage, there was one less guy holding the string.

So, a 2,500 year old technological advance cost some poor guy his job.

...sigh...

Of course what I told him doesn't sound anything like "The square on the hypotenuse is equal to the sum of the squares on the other two sides" - but what Pythagoras was getting at was this:

...and this...

The "hypotenuse" is just the longest side of a triangle that has a right angle. The "square on the hypotenuse" is that red square - and the green and purple squares are "the squares on the other two sides.

So the area of the green square plus the area of the purple square equals the area of the red square. That's all Pythagoras has been trying to tell us for the past 2,500 years.

For my house, 'a' and 'b' were two sides of the rectangle and 'c' (the hypotenuse) was the diagonal. So by multiplying the length of the house by itself, I was calculating the area of the purple square, multiplying the width by itself got me the area of the green square...and adding them together got me the area of the red square.

The "square root" button figures out what the length of the red square must be - and that's the length of the diagonal line across my house.

With all three sides of the triangle measured out in string, pulled tight and staked down - we have a perfect right angle. Then they just had to do it again with the other side - and they'd have a perfect rectangle.

MUCH easier.

Pythagoras is handy any time you need to compute a diagonal distance. You could use it to figure out the length of the roof timbers for a house too - if you know how wide the house is - and how high you want the peak of the roof - then pythagoras' theorem will tell you how long your timbers need to be.

Some people want to know how Pythagoras figured this out - to be honest, we're not absolutely sure - but probably he did something this:

You can take the bottom square and chop it into four pieces - then assemble the four pieces with the yellow square from the shorter side to make a square on the hypotenuse.

It's interesting to know that Pythagoras didn't have to pick squares. The sum of the teapot on the hypotenuse is equal to the sum of teapots on the other two sides! Any shape will do - it's just easier to use squares because calculators don't have a "teapot area" button.

From: Steve Baker | Date: 2017-10-26 12:21:04 |

I wondered about that - but the guy told me that they don't to it because the precision with which they can extend an existing line (from the 3/4/5 triangle) out to the width and length of a house on uneven ground isn't good enough.

With his original method, when they finish juggling around with tape measures and bits of string - he's confident that the final, finished corner markers are within a half inch or so of the exact rectangular shape needed.

Even with pythagoras under his belt - he still went out and measured the diagonals and four sides again at the end, just to be sure.

But, as I said - it's a true story - it really happened to me. So clearly he wasn't using the 3/4/5 method in reality.

From: Woelore | Date: 2017-10-25 10:45:13 |

That's still the hard way. Builders have long used the right-angled 3x4x5 triangle proportions to get a right angle. Set a corner, set the baseline side, measure out 4 units on the baseline from the corner and place a mark. Set a temporary second side from the corner and measure out 3 units, place a mark. Now adjust the angle of the second side so the two marks are precisely 5 units on the diagonal and you have a right angle. ONE person can do it working alone.

If you are laying out a shed for example, you might measure out 8 ft on the long side, 6 ft on the short side, and set the diagonal to 10 ft. Voila, Square.