# Constructing a 30 Degree Angle

Compass and straight edge constructions are of interest to mathematicians, not only in the field of geometry, but also in algebra. For thousands of years, beginning with the Ancient Babylonians, mathematicians were interested in the problem of "squaring the circle" (drawing a square with the same area as a circle) using a straight edge and compass. This problem is equivalent to finding the area of a circle. It turns out that this is impossible, but no-one managed to prove this until 1882!

However, it is possible to construct a 30 Degree angle using only a straight edge and compass. No cheating and using a protractor or a set square! For this construction, you will need a straight edge (ruler - but you won't be measuring anything), pair of compasses, a pencil and paper. I have drawn the pictures using the robocompass app. It's fun to play with, and you can use it to do all sorts of geometric constructions. There's a little bit of coding to learn, but a list of instructions is provided. Once you've written your little program, you can get out the popcorn, sit back and watch the construction. Be warned that, when you are using it to draw arcs, the robocompass compass point may appear to be a little away from the centre, but the drawing is actually accurate.

## The Construction

Step 1: Start out by drawing the line segment $AB$ that you want to base your angle on. The point $A$ will be the vertex of your angle. Please don't be like Sam! Use a ruler to draw this.

Step 2: Place the tip of your pair of compasses on point $A$, and open them out to a comfortable size (it doesn't really matter, but don't make it too small, or you won't have room to move). You are going to draw three arcs of this radius, so make sure you keep it fixed. Draw an arc from almost vertically above the point $A$ down to pass through the line segment at a point $D$. The arc is centred at point $A$, so make sure your compass tip is exactly on point $A$. The picture on the right shows what your construction should look like when you've finished this step.

Step 3: Keeping your compasses at the same radius as in the preceding step, place the tip of your pair of compasses on point $D$. Draw a large arc that cuts your first arc at a point $E$. The arc should be centred at point $D$, so make sure your compass tip is exactly on point $D$. As shown in the picture, your arc needs to intersect the arc you drew in step 2. The picture on the right shows what your construction should look like when you've finished this step.

Step 4: Keeping your compasses at the same radius as in the preceding step, place the tip of your pair of compasses on point $E$. Draw an arc that cuts the arc you drew in step 3. The arc should be centred at point $E$, so make sure your compass tip is exactly on point $E$. The picture on the right shows what your construction should look like when you've finished this step.

Step 5: Label the point of intersection from the previous step with a $C$. Use your ruler to draw a line segment joining points $A$ and $C$. This is the second arm of your $30^\circ$ angle. The picture on the right shows the completed construction.

### Description

This tutorial will expose you to Rulers and how to use them. We will also have a close look as set square and compass constructions. Your feedback is important to us, if you like any other topic covered under this tutorial, please do let us know.

### Audience

Year 10 or higher

### Learning Objectives

Ruler and compass constructions and more

Author: Subject Coach
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