Question

A regular hexagonal foundation 3 in. deep is being poured for a hexagonal building. If the distance across the flats of the foundation is 27 ​ft, how many cubic yards of concrete are​ needed?

Answer 1

We are given the side to side distance of the regular hexagon, and we have to find it's area
(which multiplied by the height will give us the volume of the foundation):-

Let us draw lines to divide the regular hexagon into six equilateral triangles.

Now, as the distance between across the flats is given as 27 feet, then the altitude of each of these equilateral triangles is 27/2 = 13.5 feet

Now, if h is the height of an equilateral triangle, then it's area is given by

So, as each equilateral triangle in our problem has a height of 13.5 feet, the area is given by

Now, there are 6 such equilateral triangles, so total surface area of the foundation is

Now, the depth of the foundation is 3 inches, that is, 3/12 = 1/4 feet. So the total volume of the foundation is given by

Now, we are asked to find the number of cubic yards of concrete required, so, we have to convert this to cubic yards. To do this, we just divide the whole thing by 27 to get:

Simplifying, we get the answer as:-