Quandaries
and Queries 

We are starting up a new business and for the business we will be required to order concrete for various shapes. Right now we have a job that requires a concrete slab that a decagon it is 1403/4" from flat side to flat side, 74" from point to center and each flat side is 46" The pad should be 4" thick with 12"x12" continuous footing. I remember learning some of this stuff in college algebra however I have been out of school more years then I am willing to admit. I thank you in advance for your help and look forward to hearing from you soon. Tanya 

Hi Tanya, I will calculate the volume of the slab you describe and then show you a way to estimate the volume that is easier and probably "good enough". Below is a diagram of a decagon. When you draw the diagonals you can see that it can be divided into ten isosceles triangles.
Each triangle has base 46 inches and side length of 74 inches. I drew one such triangle below.
Pythagora's Theorem applied to the triangle ABC gives AC = Sqrt[74^{2}  23^{2} ] = 70.33 inches. Thus the area of each of the ten triangles that make up the decagon is
Thus the volume of the concrete slab is
One yard is 36 inches so the volume, in cubic yards is
My suggestion for an estimation of this volume is to imagine that the concrete slab is a circle of radius 74 inches and thickness 4 inches.
The volume of this slab is
which is
The difference between 1.39 and 1.47 cubic yards is quite small and the second calculation is much easier, so I would suggest that for other slabs you use the circle approximation. By "with 12"x12" continuous footing" I assume you are describing a footing around the perimeter that is 12 inches wide and 12 inches deep. Again I am going to approximate by a circular slab.
The area of the top of the footing is the area of the outside circle minus the area of the inside circle. That is
and hence the volume is
or
It is my experience that the smallest amount of concrete you can order is half a cubic yard so you will need one and a half cubic yards for the footing and then another one and a half yards for the pad. Harley 

