if the wrench just fits the hexagonal nut what is the value of x

Answer 1

Difficult to know the value of anything, since no values are given, nor what is meant by ‘x’. As TJ did, we just make up an interesting problem, and hope that’s the one.

I assume we are given a hexagonal nut, the wrench "just fits”, we read off a value on the wrench (call this ‘m’), and we ask: What is the length ’s’ of each side of this hexagonal nut ?

A hexagon can be decomposed into 6 equilateral triangles, joined at the center of the nut.

The value ‘m’ will measure the distance from 1 side of the nut, to its opposite side… both sides parallel.

This means that the measured ‘m’ is “twice" the “height” of any of the 6 triangles of the hexagon.

Height ‘h', by symmetry, cuts an equilateral Δ into 2 congruent Right Triangles, where side ’s’ becomes a hypotenuse. The angle opposite the height ‘h’ is base angle 60º.

h = s*sin(60º)

m = 2h = 2s*sin(60º) = 2s*cos(30º) = 2s(√3 / 2) ==> m = s√3 ==>

If we are given m, then find s:

s = m / √3 = (√3 / 3)m

Note: TJ and I came with different values which are flip-sides of the same problem.

Cool fact: The “size” of a nut is the measured distance “across flats”, so our m, and TJ’s x.

(a) In TJ’s case, we are given s, and must find the “size” x = s√3 (like above)

(b) In my case, we are given measured size m, and find s = m / √3 = (√3 / 3)m

Reminder:

Easy to forget...In a 30º-60º-90º triangle, sin() of the smaller side = 1/2, and smaller side (in this case) is opposite the smaller angle 30º, so sin(30º) = 1/2. With a side of 1/2 and hypotenuse of 1, the missing side opposite the 60º angle, by Pythagorean theorem is:

√(1² - (1/2)² = √(1 - 1/4) = √(3/4) = (1/2)√3 or √3 / 2

Answer 2

let each side of the hexagon is of length s

then

x = 2s * cos(30°) = 2s * (√(3)/2) = s√3

now, in order to find the exact value of x we need the length (s) of hexagon.