## 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.

ur a nerd

joe said this on August 24, 2007 at 10:29 am

very interesting.

i’m adding in RSS Reader

music said this on January 8, 2008 at 6:49 pm

goodshit! needed it 4 class!

Christian said this on April 22, 2008 at 5:58 am

excellent. putting this to use tonight.

tim said this on May 30, 2008 at 12:11 pm

My garden will be square thanks to you.

Fred said this on September 22, 2008 at 6:54 pm

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..

Joe said this on October 30, 2008 at 4:57 pm

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

annie said this on July 20, 2009 at 4:01 am

[…] 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. […]

Terrys multi-level low level Deck step by step, work in progress - Renovate Forums said this on November 16, 2009 at 4:30 am

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

jewel thompson said this on April 27, 2010 at 4:19 pm

this is owesum

Domingo wakalenda said this on October 27, 2010 at 11:56 am

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

CDH Renovations said this on March 1, 2011 at 9:27 am

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

msrebekahjane said this on September 14, 2011 at 8:07 am

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

Bob said this on October 21, 2011 at 10:28 pm

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

H3 Maths - Jesus and the 345 Triangle Rule of Pythagoras said this on March 28, 2012 at 8:02 pm

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

Ryan Banman said this on August 15, 2012 at 8:00 pm

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.

edmundo said this on December 23, 2012 at 12:24 pm

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.

Andy said this on February 22, 2013 at 3:21 pm

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!

Alan Woodstock said this on March 7, 2013 at 3:34 pm

This is a topic that is close to my heart… Best wishes!

Where are your contact details though?

Arlie said this on April 5, 2013 at 5:14 am

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!!!

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walkfit platinum reviews said this on June 21, 2013 at 9:22 am

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.

Craig Siefkas said this on August 13, 2013 at 5:34 pm

sweet! now math is a lot easier!

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bobby pants said this on October 28, 2015 at 10:08 am

ithis helped me a lot

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porridge vulture said this on November 12, 2015 at 8:01 am

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.

Kathy Blair said this on February 7, 2016 at 6:27 pm

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nerdkiller said this on February 19, 2016 at 8:23 am

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.

END2008 (@END2008) said this on May 18, 2016 at 6:46 am

Gotta love the Pythagorean theorem

Shane said this on July 14, 2016 at 12:36 pm

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

Lee said this on August 5, 2016 at 3:54 am

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

Jeanne said this on August 5, 2016 at 5:50 am