Did you know that you can make your own protractor at home by folding a square piece of paper. We know that a square piece of paper has four 90° angles and by folding it in various ways we can divide the 90° angle into equally smaller angles. To begin with, you can bisect a right angle to form a 45° angle by folding the paper diagonally. But how do you find a 30 degree angle without a protractor or a 60° , 120°, 75°, and so on ? Let’s try to work that out.

## What can You use if YOU don’t have a protractor

- A square piece of paper
- a Protractor for testing the paper protractor ( Optional )
- a pen

## How do you find an angle without a protractor

Take a square piece of paper and fold it in half and then unfold it.

Pick up the upper left corner ( with an angle of 90° ) and fold so that the corner point touches the mid line you created in step 1, and move it such that the creased line passes through the top right corner of the square. This fold will create a 30-60-90 triangle.

Now fold the bottom left corner so that the fold being made lines up with the edge of the already folded over 30-60-90 triangle. It will create the second 30-60-90 triangle. Tuck it under the bigger 30-60-90 triangle you created in step 2.

Tuck it under the bigger 30-60-90 triangle you created in step 2 and mark the angles.

Fold the paper so that the remaining unfolded top right corner is bisected in such a way that the bottom right corner meets the edge of the 30-60-90 triangle you formed in step 2.

You protractor is ready and you can now find and mark all the angles on it.

Open the paper and you can see that one edge of the paper has been divided into three equal angles of 60° each.

## MATHEMATICAL PROOF

So what’s going on? How can you prove that the right angle triangle formed in step1 and step 2 are 30-60-90 triangle ?

Let’s assume the length of the square side is 2a. The line A’B aligns with the side AB of the square when the fold is made, so it is also 2a.

In the right angled triangle A’EB, length of side BE is ‘a’ and the hypotenuse is ‘2a’. therefore it is a special right angle triangle. Hence, the angle EA’B is 30° and the other angle EBA’ in the triangle must be 60°.

Now, in the quadrilateral AE’A’B, we already know that some of the angles are 90° and 60° . Angles E’AB = 90°, angle E’A’B = 90° and angle ABA’ = 60°. Since the angle sum of a quadrilateral is 360° and the fold line BE’ is a line of symmetry, the missing two angles ( angle AE’B and angle E’A’B ) must each be 60°.

That’s how we can prove that the triangle created in the first fold is a 30-60-90 triangle.

RELATED POST : How to prove that the sum of all angles of a triangle is 180 degrees

The mathematical magic of demonstrating geometry principles and constructing geometric figures using paper-folding doesn’t end here. There are many books written on it.

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I hope your kids will enjoying understanding how to make a protractor using paper and learn the basic concepts of geometry just by folding a piece of paper.