The sides of the regular hexagonal nut shown below are x = 17 millimeters long. Find the height h of the nut.

Question

The sides of the regular hexagonal nut shown below are x = 17 millimeters long. Find the height h of the nut.

 

 

Answer 1

draw a diagram

draw the apothem, the distance from the center of the nut perpendicular to one side

from the center draw a line to the adjacent vertex (either side)

you have a right triangle

360/60=60º

60/2=30º

this is the angle at the center of the nut formed by the apothem and the hypotenuse

tan=opp/adj

tan(30º)=8.5/a where 8.5 is half of the side of the nut and "a" is the apothem

a=8.5/tan(30º)

tan(30º)=√3/3

a=8.5/(√3/3)

a=[3*8.5/√3)

a=25.5/1.732

a=14.72286

double the apothem to get the height of the nut

2*14.72286=29.4457 mm

 

Answer 2

Draw and label a diagram!

Draw the diagonals of the hexagon - six equilateral triangles are formed.

What follow depends on what is meant by "height."

If it is from vertex to vertex: 2(17mm) = 34 mm

If it is from side to side: 2(17√3/2) = 17√3 mm    (30-60-90 triangle!)