## 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.

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:

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:

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:

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.

~ by mikelindstrom on January 22, 2007.

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

1. ur a nerd

2. very interesting.

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!

24. llama

25. OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOps

26. ithis helped me a lot

27. SNAPE SNAPE SEVERUS SNAPE

28. I do not understand why both legs can’t be same length and then then draw a line from those points. Isn’t this a 90-degree angle.

29. F***ing NERDS!!!!! get some vagina virginsss!!!!!!!!!

30. The math is straight forward , i.e 3,4 5 triangle however that only tells you the math.IT does not tell you how for example to deal with a wall / room you find is not square.

31. Gotta love the Pythagorean theorem

32. Used in roofing, chalk lines on the roof. All ya need is a straight line acrostic bottom to go off of.

33. Thank you, just what I was looking for. The elaboration was helpful as well. jyd

34. is there a tool 6-8-10 square? If so where can I get one from

35. I have not checked in here for some time because I thought it was getting boring, but the last handful of posts are really good quality so I guess I will add you back to my daily bloglist. You deserve it my friend.

36. Great explanation ! Forgot this rule and needed a perpendicular line for my new columns.

37. I just used it on a small scale 3″ 4″ 5″ on my coffee table and worked. Check you inner diameter (y) & (x) length and width to assure it’s to your measurements, then square the corners to assure squarness. Adding string lines on bigger project are useful with property marking for say if laying a pad of a house.

38. I’m trying to get a 105 degree angle in the field, as opposed to a 90, Do you know what measurements I would pull to get the hypotenuse?

39. i know i am missing something my square is 8 ft x 8ft so 8×8 = 64 64 + 64 =128 ft corner to corner ?