Question

Write the distance y across the flat sides of a hexagonal nut as a function of r (see figure in the website below).
www.webassign.net/larpcalclim2/4-8-049.gif
y=

Answer 1

Enclose the hexagon within a square
Consider the right-angled triangle at the top right.
It is a 30-60-90° triangle with a hypotenuse of r
(The sides of this triangle are in the ratio of 1 : √3 : 2 )

let the side opposite the 60° angle be p
p/r = √3/2
p = √3r/2
but y = 2p = √3r