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.