Cohn's theorem

Okay kiddo, have you ever played with a jigsaw puzzle and tried to put all the pieces together to create a picture? Well, mathematicians also like to put together pieces to create a bigger picture.

One thing they like to do is work with shapes called polygons, which are just flat figures made up of straight lines. Now, sometimes we need to measure some things about these polygons, like their angles or sides.

That's where Cohn's theorem comes in. This theorem helps us find out the lengths of the sides of a special type of polygon called a regular polygon, where all the sides are the same length and all the angles are the same size.

Basically, Cohn's theorem says that if you have a regular polygon with n sides (let's just use the number 5 for now), you can find out the length of one of the sides by multiplying the radius of a circle circumscribed around the polygon (that means a circle that just touches all the corners of the polygon) by 2 times the sine of 180 degrees divided by n.

Okay, I know that might sound a bit confusing, but let me give you an example. Imagine we have a regular pentagon (a polygon with 5 sides) and we want to find out how long one of the sides is. We can draw a circle around the pentagon that just touches all 5 corners, and measure the radius of that circle. Let's say the radius is 5 centimeters.

Now we can use Cohn's theorem to find the length of one of the sides. We first divide 180 degrees by 5 (since we have a pentagon) to get 36 degrees. We then take the sine of 36 degrees (which is about 0.5878). We multiply that by 2 and by the radius of the circle (which is 5). This gives us a final answer of about 5.877 centimeters.

So there you have it, Cohn's theorem helps us find the length of the sides of regular polygons. Pretty cool, huh?