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Geometry - Circles 2

An article by Andy Slater with illustrations by Aris Kafantaris.

So you rescued the princess (see previous instalment) and grateful though she is, she's not willing to kiss you just yet and wants you to go back the the stone circle, divide it up into 12 segments (one for each sign of the zodiac) and turn it into a flower garden in honour of her aged mother.

Now you should see this as a clear indication that any future that you may have together will be and endless succession of DIY projects and household chores, but you're young and in love and you need to snog her if you're ever going to get rid of that turn into a frog during the hours of darkness thing that's been irritating you since your encounter with the witch. Thus you pay a quick visit to the garden centre and head for the enchanted forest.

Now, as you are able to find the centre of any circle with relative ease, we have a starting point, but how about splitting that circle it into segments?

2, 4, 8

I'm not going to insult you by telling you how to split a circle with a known centre into 2. As for splitting it into 4, if you know how to measure right angles then splitting your circle into 4 is easy. Splitting it into eight is a little more difficult but if you also know how to bisect a line you can do it as illustrated by steps 3 and 4 in the image below.

Geometry - Circles 2

In a nutshell, you draw lines between the points on the circumference and bisect them. The same procedure can be used to split the circle into 16, 32, 64, etc. however none of these will give us the 12 that our new love requires.

3, 6, 12

There is a technique for splitting a circle into 6 using a pair of compasses (and in the forest we can use a rope) if you can find the centre, and of course, we can.

To split the circle into 6, simply set the compasses to the radius of the circle. Now choose a point on the circumference and put the pointy end of the compasses on your chosen point and mark two more points on the circumference by rotating the compasses. Now move the point to of the compasses to one of your marks and mark a further 2 points. You will notice that one of these marks is over the place where the point of the compasses was last time. Continue around the circle until you have 6 points marked around the circumference.

Geometry - Circles 2

Joining these up the six points with three lines (each of which passes through the centre) divides our circle into 6 segments. If we only wanted 3 then we could choose just three of our 6 points and join them at the centre but we want 12. The trick is to join up each of the 6 points to make a hexagon and employ the same bisecting technique that we used to turn 4 segments into 8, to turn our 6 segments into 12.

Pucker up princess, it's kissing time.

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