Getting Things Square With the World: 3-4-5 Triangles

Construction projects often need to have precise 90 degree or “square” angles. But often the available tools (such as a carpenter’s square) are simply too small to guarantee the accuracy needed for large projects such as laying out the foundation of a house. So carpenters and concrete formers will often employ a 3-4-5 triangle technique to ensure accurate 90 degree angles.
3-4-5-triangle-with-lines.jpg

The technique simply requires that a person create a triangle in the corner of the lines that are to be square (90 degrees) to each other. The triangle must have one side (leg) that is 3 feet long, a second side that is 4 feet long and a third side that is 5 feet long. Any triangle with sides of 3, 4 and 5 feet will have a 90 degree angle opposite the 5 foot side. If a larger triangle is needed to increase accuracy of very large structures, any multiple of 3-4-5 could be used (such as a 6-8-10 foot triangle or a 9-12-15 foot triangle).

So, mathematically why does this technique create a perfect right angle??

In Geometry, a well known method of constructing a right angle is to employ the Pythagorean Theorem. The mathematician, Pythagoras, discovered a relationship between the sides of any right triangle that is now known as the Pythagorean Theorem; he proved that the square of the longest side (the hypotenuse) is equal to the sum of the squares of the remaining two sides. This is often expressed as the following equation:

abc-triangle.jpg

a2b2c2.png

where A and B are the two legs of the right triangle and C is the hypotenuse. If we substitute the numbers from a 3-4-5 triangle into this formula, we then have:

324252.png

Of course any lengths could be used to create the right angle for construction – as long as they were correct when applied to the Pythagorean theorem. But practically speaking, most other numbers would not work well. First of all, finding a square root on the job site would often require a person to carry a calculator around. Secondly, once a square root was found, it often could not be accurately located on a tape-measure or other measuring tool.

For example; suppose the carpenter chose to use triangle legs of 6 and 7 feet. Using the Pythagorean theorem we would find:

6272c2.png

Solving for C would produce a hypotenuse of 9.2195444 feet – which would be very difficult to calculate mentally or locate on a tape-measure that is graduated in 8ths or 16ths of an inch.

Therefore using triangle dimensions of 3, 4 and 5 is easy to remember (no calculations needed), will always produce a perfect right angle and is easily found with common measuring tools.

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~ by mikelindstrom on January 22, 2007.

23 Responses to “Getting Things Square With the World: 3-4-5 Triangles”

  1. ur a nerd

  2. very interesting.
    i’m adding in RSS Reader

  3. goodshit! needed it 4 class!

  4. excellent. putting this to use tonight.

  5. My garden will be square thanks to you.

  6. I tried the 3 4 5 method to get square today but it just didn’t work. I don’t know what I did wrong. I am going to try again tomorrow..Thanks for the info..

  7. thank you….so simple, but was not sure of the jargon, using info tomorrow, thanks again.

  8. […] where posts are not required. Next we used packers to set the bearers at the right height and the 3-4-5 method to make sure everything is square. Here we are packing up the bearers and putting them in place. […]

  9. i need to make aight angle triangle with one leg 6 feet long, how do i do it?

  10. this is owesum

  11. This was the fastest way to show my employee how to square up walls. So well explained. Thank you.

  12. Thanks, will use today in setting posts for my grape arbor. :-}

  13. Thank you for this! I’m installing flooring tomorrow.

  14. […] from the Pythagoras theorem for right-angled triangles?” Most carpenters and home handymen use this regularly to square walls, etc. What do you […]

  15. Hey I”m not real good at math, but wouldn’t the nonuseful axample work if you used the large numbers to read the inches on the tape? In many cases, at least with my tape, it’s easier and faster to read the inches than the foot markings.
    So say you use 6’x7′ legs, turn it into 36″x47″ and then measure 85″ on the 3rd leg.
    I haven’t tried this, I’m just theorizing :)

  16. I’m a carpenter and this information is gold, essential to my trade to know how to square a structure without having to use a square. thank you for the info its gold.

  17. Mike–Came across this post when I googled a sentence my student had in his Geometry extension project. Kind of a funny coincidence that he plagiarized someone I know! Hope you are doing well.

  18. The so called carpenters who had to come on here to find this out hang up yer nail bag an do something else an leave it to the pro’s
    Haha this is 1st yr carpentry & joinery level 1 stuff here
    As for everybody else I was actually taught this in secondary school, and btw my school was a shithole in glasgow’s east end where going to school and getting murdred in the playground was a daily possibility!

  19. This is a topic that is close to my heart… Best wishes!
    Where are your contact details though?

  20. I am a cabinet maker and 20 years ago we wood use this regularly, today with all the computer assist software we are getting LAZY. Thanks for the post!!!

  21. We appreciate you the excellent writeup. Them the simple truth is was previously a new fun bank account it. Appear advanced to help much supplied flexible from you! In addition, precisely how may possibly most people keep up some sort of distance learning?

  22. Have to laugh, the Bullitt County History posted an eighth grade test from 1912. (http://www.bullittcountyhistory.com/bchistory/schoolexam1912.html) Question 8 in mathematics was: bingo! the 3 4 5 rule. Actually the answer was the 3 4 5 rule. Used this for many a shed and barn. Only we used the 30 40 50 rule, because I’ve got a 50 foot tape measure.

  23. sweet! now math is a lot easier!

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