   SEARCH HOME Math Central Quandaries & Queries  Question from Teresa: How many cubic yards of concrete are required for the structure with a bottom piece that is 8" high by 72" by 72", a side that is 8" by 72" by 60", another side that is 8" by 72" by 60 ", a side that is 8" by 72" by 60" with a circle cut out that has a radius of 18", and finally another side that is 8" by 72" by 60" that has a circle cut out that has a radius of 12". Teresa,

All the slabs are 8" deep by 72" wide. If you add the lengths together, it is 72+60+60+60+60 = 312". The volume, if you didn't cut any circles is 8 x 72 x 312 = 179712 cubic inches.

From this, you would subtract the volume removed for the holes. These holes are cylinders and the volume of a cylinder is the depth times the area of the circle, which is pi times the radius squared. There are two holes with two different radii. So the volume to remove is 8 x pi x 182 + 8 x pi x 122 = 11762 cubic inches.

Therefore the volume of the concrete is 179712 - 11762 = 167950 cubic inches.

To convert this to cubic yards, remember two things: (a) a cubic yard is (1 yard)3; (b) 1 yard = 36 inches.

Therefore, (1 yard)3 = (36 inches)3 = 363 inches3 = 46656 cubic inches.

So divide by this to get the number of cubic yards:

167950 cubic inches / (46656 cubic inches per cubic yard) = 3.6 cubic yards.

Cheers,
Stephen La Rocque.     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.