6
$\begingroup$

I am an engineer in the oil field and I am trying to find a socket that can fit around a square drive pin. I am trying to prove this for fun before I just draw it on CAD and measure it. I believe sockets have six or twelve sides. I have had some success calculating the largest hexagonal wrench that can fit on a square drive, but it came out very large and I remember that it was not this large of a tool when I worked in the field. The problem is that when you add this extra point in a hexagon making it a decagon, it allows the square to rotate. The square drive is d=.750 and I came out with x=(sqrt(5)/(3-sqrt(3))d, where x is the width across two sides of a hexagon. The proof of the hexagon starts with the fact that the largest square inside of a hexagon shares the same center. Its a strange proof that I dont understand, but it can be found here: http://www.drking.org.uk/hexagons/misc/deriv3.html.

This web page gives us the result that (1) d=(3-sqrt(3))a. Then I derived (2) a=x/sqrt(5) using the Pythagorean therom. From there, I substituted (2) into (1). and got the result x~1.764d, x~1.323 when d=.750.

I know this cant be right. I could calculate the diagonals of the square drive and set that equal to the diagonals of the dodecagon. From there I could probably find the distance between two sides of the dodecagon and that would be a starting point, but I'd rather find a relation and proof for largest square inside a dodecagon that cannot rotate. I know aligning the square diagonals with the dodecagon diagonals would not be correct because I do not think this is how you'd want to start this proof.enter image description here

$\endgroup$
  • $\begingroup$ @john wayland bales I overlooked the difference that adding another 6 sides would make. It looks correct. $\endgroup$ – Mike K Apr 30 at 2:02
2
$\begingroup$

Wouldn't it be easier with a regular dodecagon, since 4 divides evenly into 12?

The largest square which will not rotate will fit snugly into four corners of the dodecagon.

Square inscribed in dodecagon

Note that in $\triangle AOM$, side $OM$ must be the largest side. Since $\angle MOA<30^\circ$ then $\angle MAO>75^\circ$.

Proof in pictures

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.