The Geometry Of A Star

Please help my flu-addled brain…

Picture a standard, five-point star. If the length of each line were 36", then how long would a direct line be if it went from the topmost point to the rightmost point?

Using the cosine rule it’s sqrt (36^2 - 236^2cos 36) (The last 36 is the internal angle, not the lenght.)

You are basically describing a pentagon with a star drawn inside of it. The pentagon diagonal is 36" and you are looking for the length of the side of the pentagon. The ratio of the diagonal to the side is the “golden ratio” and it works out to (1+ sqrt(5) ) /2, or about 1.62.

So if the diagonal is 36" then the side is 36/1.62 or about 22.2".

Derivation of the formula here (showing a nice star/pentagon picture)

Wikipedia article about the golden ratio:

22 1/4" works for me – much appreciated!

Using the cosine rule correctly it’s sqrt (236^2 - 236^2*cos 36)

Which is 22.25. Just in case someone other than me tries to check my numbers and gets as confused as I did.

I’m surprised that the cosine rule, etc. gets mentioned when this is so obvious. First thing that came to my mind. Plus I remember more digits, 1.618. So a quick Google calc with a single division and you have 4 digits. (Or you could multiply by 0.618.)

When I plugged them into Google, it also returned web pages such as this one (look at the end of article), that needed the same ratio, but usually for other reasons.