How to make a Protractor with Paper

Did you know that you can make your own protractor by folding a square piece of paper. We know that folding a paper in different ways can divide an angle into equally smaller angles. To begin with, a square piece of paper has four 90° angles and we can bisect a right angle to form an angle of 45° by folding the paper diagonally. But what about different angles like 30°, 60° , 120°, 75°. Let’s try to figure it out.

Materials

• A square piece of paper
• a Protractor for testing the paper protractor
• a pen

Instructions

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.

Further Explanations

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.

I hope your kids will give it a try and learn how fun it can be to learn the basic concepts of geometry just by folding a piece of paper.

While you are here, check my post to prove that the sum of all angles of a triangle is 180° by simply folding a piece of paper HERE.

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