hexagonal volume

Question

You have been brought a hexagonal vase from a crime scene that is filled with blood to 12.0cm above where it rests on the table. Each side of the vase is 5.00cm (measured from the outside surface) and the distance between one outside corner and the other directly across from it (going through the vase) is 10.0cm. The base of the base is 0.500cm thick and the walls are 0.250cm thick at each corner. What is the volume of blood contained in the vase?

Answer 1

We have to find the volume of the inner hexagonal prism. First, each side will be reduced by 2Δs, where Δs comes from looking at one corner, where the hypotenuse of a 30-60-90 triangle is 0.25 cm (thickness AT the corner) and Δs is opposite the 30° angle, so Δs = 0.25/2. Internal side length is 5.00 - 0.25 and perimeter P = 6 * 4.75 = 28.5 cm. Apothem a comes from another such triangle as one of 6 identical sectors of the hexagon, where hypotenuse = (10.0 - 0.25*2) /2 (half of diagonal) and a = altitude = (√3 /2) (hyp = 4.75) = 4.11₄ cm.

Area of inner hexagon A = 1/2 P a = 58.6 cm². For volume we multiply this by interior height h = 12.0 -0.5 = 11.5 cm to get V = 674 cm³.