If you were awake when your high school math teacher was explaining the Pythagorean theorem, that knowledge and a tape measure are all you need to lay out an accurate 90-degree angle. Even if you missed class that day, it's still easy to understand the technique, which comes in handy for laying out new structures as well as checking the square on existing ones.
Right-Angle Measurement Using the Pythagorean Theorem
The key to laying out a perfect 90-degree angle is to construct a right triangle, which is one with one 90-degree angle. According to the Pythagorean theorem, the lengths of the sides of any right triangle (a, b and c) are related by the expression:
a2 + b2 = c2
Now suppose the length of side "a" is 3 units and that of side "b" is 4 units. If you plug those numbers into the equation and solve, you'll find the length of side "c" to be 5 units.
The 3-4-5 method works for any values of "a" and "b" as long as you can reduce them to a 3:4 ratio. For example, if "a" is 6 and "b" is 8, then "c" is 10, and if "a" is 33 and "b" is 44, then "c" is 55. This is good to know when you have to change units.
How to Use the 3-4-5 Rule
Suppose you want to construct a fence, and you've set the first corner post. You want to be sure the lines that extend from that post form a 90-degree angle at the post. Here's how to do it:
- Draw a chalk line or stretch a string in the direction of one side of the fence. Measure 3 feet along that line with a tape measure and make a mark.
- Draw another line in the general direction of the other side of the fence and make a mark at the 4-foot point on that line.
- Extend the tape measure between the marks. Without changing its distance from the post, adjust the position of the second mark until it is exactly 5 feet distant from the first. The angle between the fence lines is now exactly 90 degrees.
If you don't have string or chalk, you can still employ this method using only your tape measure. Just extend the tape out and make marks on the ground at the appropriate distances from the post.
Checking Square With the 3-4-5 Method
The Pythagorean theorem comes in handy when you're framing walls, hanging doors or building cabinets. One way to ensure the angle between two sides is 90 degrees is to check it with a framing square, but you can also mark 3 units on one side and 4 units on the other and then measure the distance between them to ensure it's 5 units.
Contractors use a variation of the 3-4-5 method to check the square of door openings. They measure the distance from one top corner to the diagonally opposite bottom corner and compare this to the opposite diagonal. Given that the two sides of the frame are the same height and the top and bottom of the frame are the same length, the diagonal distances should be the same.
If they aren't, the opening must be out of square, and that usually means one of the sides isn't plumb. To determine which side, measure 3 units along the top, 4 along each side, make marks and then measure the distance between the marks. The side that does not measure 5 units between the marks is the one that needs adjusting.