Hi Doug and Andrea,
I am assuming the measurements are in inches.
The volume of your tank is the area of the base times the height. Since the base of your tank is a hexagon you need to find the area of a hexagon. To do this draw a hexagon and join the opposite vertices to divide the hexagon into 6 congruent triangles. Thus the area of the hexagon is 6 times the area of one of the triangles.
The sum of the 6 angles at the centre of the hexagon is 360^{o} and hence each of the 6 angles measures 60^{o}. By the symmetry, each of the other angles of the triangles measure 60^{o} and hence each is an equilateral triangle with sides of length 14 inches. Look at the bottom triangle.
Draw a perpendicular from the apex to the base and call its length h inches. You then have a right triangle with sides of length h, t and 14 inches and thus, using Pythagoras theorem,
h^{2} + 7^{2} = 14^{2}
and thus
h^{2} = 14^{2}  7^{2} which gives h = 7 √3 inches which is approximately 12.12 inches
The area of a triangle is (half the base) height and hence the area of each of the 6 triangles that make up the hexagon is
7 7 √3 = 84.87 square inches.
Thus the area of the hexagon is
6 7 7 √3 which is 509.22 square inches.
Finally the volume of the tank is
(area pf the base) height = 6 7 7 √3 29 = 14767.5 cubic inches.
There are 231 cubic inches in an American gallon and thus the volume of the tank can also be expressed as
^{14767.5}/_{231} = 64 gallons
Penny
