The following instructable details 2 ways to find out if something is square, 1 way to draw an accurate perpendicular line, and 1 way to draw an accurate parallel line.
These tricks involve virtually no math to do and are scalable to any dimensions from millimetres, to miles.
The 4 tricks are probably not something you will use every day but are really handy to know and will allow you to amaze others with your skill and mental prowess.
I have embedded the video podcast of this because it is sometimes easier to see it done than to try to understand a written description. my part is at 5:24 Don't be shy about watching my other episodes or checking out my other instructables.
Step 1: Trick #1 - the 3,4,5 Trick
Note* this trick is the only one that uses math and it is only really to explain how it works. So don't be scared off.
According to Pythagoras, a right angle triangles sides can be described by the equation
a squared + b squared = c squared
where c= the hypotenuse (the longest side).
By a lucky fluke of math 3 squared + 4 squared just happens to = 5 squared. So all you have to remember is 3,4,5. Simple so far right?
If you have something you are working on, for example a wooden frame, cut plywood, a tacked metal structure or even brickwork, and you want to see if it is square, this trick is for you. Just measure 3 units across from the corner and make a mark, then 4 units up and make a mark. Now measure the distance between the two marks and you should get 5 units. If not then it is not square. If it is more than 5 units it is more than 90 degrees. If it is less than 5 units it is less than 90 degrees. I am saying units because it does not matter what you use to measure, what matters is that the 3,4,5 ratio is correct. Just know that the longer measurements you can take, the more accurate it will be. You can even do multiples of the ratio, for example 12”x16”x20”.
Step 2: Trick #2 - the Equal Diagonals Trick
This method is a little easier than the 3,4,5 trick but requires access to all 4 corners of a rectangle or square. This would work great for a picture frame. Just measure across the diagonals from corner to corner. If the measurements are equal then the corners are square.
Step 3: Trick #3 - 90 Degree Arc Trick
This is great for marking things like fences or deck foundations, I use this at work sometimes to layout and locate machinery in factories. We assume you already have a straight line to go off of, like an existing wall or a reference line on the ground.
You need a piece of string to do this one, mark where you want your new perpendicular line to come out of the reference line or wall. Using the string, tie a knot about 1/3 or so from the end. Measure with the string out along the reference line both ways and make marks at the knot. From each of these two marksand using the whole string, draw an arcs like in the picture. Where the two marks intersect, that is perpendicular to your reference line at the point you started from.
Step 4: Trick #4 - the Parallel Arcs Trick
This has the same uses as trick number 3 but is used to make parallel lines instead of perpendicular. From any 2 points along the reference line (the further apart the better) make an arc with the radius of the desired offset. If you pulled a string, layed a straight edge or shot a laser line that just touched (tangent to) both arcs, it would be parallel to your reference line.