Compass and straight edge constructions are of interest to mathematicians, not only in the field of geometry, but also in algebra. Here's one that I've actually used for practical purposes. I made a quilt with this pattern (called Grandma's Flower Garden) on it. To make the blocks, I needed to make lots of templates shaped like hexagons, and I used this construction to create them.

In this article, I'll show you how to construct a hexagon inscribed in a circle 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.

Step 1: Start out with the circle (with its centre marked) that will circumscribe your hexagon. If you don't know where the centre is, don't guess! Read the article on constructing the centre of a circle, and follow the steps to find the centre of your circle.

Step 2: Mark a point \(P\) on the circumference of the circle. It doesn't matter where. This is the first vertex of your hexagon.

Step 3: Open your compasses out to the radius of the circle (the distance between the centre and point \(P\)). With the tip of the compasses exactly on point \(P\), draw an arc cutting the circle. Move the tip of the compasses up to the point of intersection of the arc you've just drawn with the circle. Draw another arc of the same radius, cutting the circle. Repeat this process until you have a total of 5 arcs cutting the circle. Together with point \(P\), the points of intersection of these arcs with the circle form the vertices of the hexagon.

Step 4: Using the straight edge (did you hear me, Sam?), join point \(P\) to the point of intersection of the first arc with the circle. Then join this point of intersection to the point of intersection of the next arc with your circle. Continue in this manner until you have a completed hexagon in front of you. Sit back and admire your handiwork. If you're making a quilt, you'll need to make quite a few of these. Have fun!