Hi Lewis,
I am going to look at a hexagon with side length L units. If you join each vertex of a hexagon to the centre you can see that the hexagon can be divided into 6 congruent, equilateral triangles.
Now look at one of te triangles and label its vertices A, B and C.
D is the midpoint of BC and triangle ADC is a right triangle, so using Pythagoras theorem we get
L^{2} = (L/2)^{2} + AD^{2}
and hence
AD^{2} = (3 L^{2}) /4
and thus
AD = (√3 L) /2
Thus the area of the triangle ABC is
(half the base) height = (L/2) (√3 L) /2 = (√3 L^{2}) /4
The hexagon is composed of 6 of these triangles so 6 times the triangle area gives
hexagon area = ( ^{3 √3}/2) L^{2} square units
The volume of your hexagonal slab is the area of the hexagon times the thickness. Since you want the volume in cubic yards you need to change the dimensions to yards before you calculate the volume. There are 3 feet in a yard so your form is 4 yards on the side and 1 yard deep. Thus the volume is
(^{3 √3}/2) 4^{2} 1 = 41.6 cubic yards.
Penny
