How can I draw right angles with just a ruler?

Question

If I am trying to draw a rectangle or square on a piece of paper I can use existing corners of the paper as a start point. However I lose accuracy when completing the square or shape on the inside of the paper. In some cases I plan on cutting this and using it a template.

I measure corner to corner to verify accuracy and I usually have to settle for good enough.

If I had a square (ruler) I don't think I would be having an issue or at least not be adjusting my lines as often.

Using just a plain ruler and pencil is it possible to draw a perfect or near perfect square or rectangle? i.e. set of 4 right angles.

Answer 1

You can use the 3-4-5 method of creating right angles.

Begin with a construction line and mark a zero point and a 3 point. Make a mark. Notice that I'm not using units of measure. You can use millimeters (preferred) or inches or anything in between.

Draw from the zero point at a right angle as closely as possible. Measure from the zero point to a 4 point. Make a mark.

Measure from one of the marks to the other and it should be a 5 point. If so, you have your right angle.

If not, you will have to erase or otherwise ignore the second line and create yet another.

I prefer using metric measurements for this type of project. You can adjust the numbers to fit your workspace. Let's say that you have a small piece of paper on which you wish to create your square.

The first line could be 90 mm long (3 * 30), while the second line would be 120 mm (4 * 30) and the points should be 150 mm apart (5 * 30).

Instead of drawing the second line as described above, consider to have a piece of paper with the length necessary for the conditions (120 mm) and position it with the zero point on the first line. Move the paper in an arc until it meets with your 150 mm point on the ruler. You can then mark the base paper with your points for a perfect square without having to erase misplaced lines.

As noted in the image below, greater accuracy is gained with larger distances. The quality of the measuring instrument also plays a factor, but only a minor part.

Answer 2

1. Draw top and bottom of the ruler to create parallel lines.
2. Rotate the ruler and repeat using the original lines to create a parallelogram.
3. Draw the diagonals of the the parallelogram, thus creating a right angle at the centre.
4. Continue drawing parallel lines with same width using the ruler and adding the diagonals.
5. A square will present itself. .

Answer 3

This is not possible using only a ruler. The only way I know of to do this in the physical universe as absolutely perfectly as possible is to use a compass and a straightedge. This method requires no measuring, but only accuracy.

1. First draw a circle, making sure to leave a mark at the center.
2. Pick any point on the circle as the position for the center of a circle with the exact same radius.
3. Draw a line through these two radii and use the intersection of line and circle to place the center for the third circle and draw one there.
4. From the center outward, draw lines through the intersections of conjoining circles. Where these lines meet, draw a line down the center to construct your perpendicular and solve for the bisection of the circle.
5. Connect the dots.

Image is my own work. CC-BY

Answer 4

If the question is really the math challenge in the title, and the explanation about drawing rectangles was just to make this on-topic here, you already have some good answers. However, if the objective is actually drawing good rectangles, and the mention of right angles is just clarification of the problem, there's a simple approach that hasn't been mentioned yet.

The paper has perfect right angles at the corners and you can take advantage of that. If you measure only along the edges of the paper, you don't have to worry about error due to the ruler not being perpendicular. Use the ruler just for measuring and drawing straight lines.

Say worst case is you need a rectangle somewhere in the middle of the page (if it is at a corner, that saves some work):

X and Y are the dimensions of the rectangle needed, and A and B are the distances from a corner. Use the ruler to measure those distances along each edge of the paper and mark those locations:

Align the ruler with the matching marks on opposite sides of the sheet and draw the connecting lines (clean up the lines beyond what's needed:

If you were precise, you've got your perfect rectangle with accurate right angles.

If the rectangle needs to be in an arbitrary rotation on the page, do the exercise above on another sheet of paper, but start the rectangle in a corner of the sheet. There will be only X marks top and bottom, and Y marks on the sides, with one horizontal and one vertical connecting line.

That will leave excess paper on just two sides of the rectangle. Use the lines that run across the sheet as guides, align the sheet edges, and fold back the excess paper, making a sharp crease on the two folds.

You now have a rectangle to use as a template. Put it in the location and orientation required on the "good" sheet and trace it.

Answer 5

If you're planning on cutting anyway fold the paper. For squares you can measure along your straight edges and then fold up a triangle using your 2 marks for your corners. You can then trace the edge of the paper to get the other sides of the square.

Rectangles you can fold each side.