geometry proof and proven

Question

The management of Faculty of Science and Mathematics intends to build a simple garden pond at the car park area in front of Block 1 building. The pond would be in the form of a regular hexagonal prism with the length of each base edge is 1.5m and an apothem of length 1.4m. The depth of the pond is 0.5m. They plan to fill up the entire pond with water. Prove formally that the volume of the pond is 3.15m^3 .

Answer 1

The problem can be solved without the knowledge of the length of the apothem, since the base of the prism

is a regular hexagon which can be partitioned to 6 equilateral triangles. If we let s be the side of one of the 6 equilateral triangles then the height of each of the 6 equilateral triangles is H = (s√3)/2 = 1.5⋅√3⋅(.5) ≈1.29904m applying the Pythagorean Theorem in the right triangle with sides ( s/2 , H , s)

.Now is the moment of truth since the height H is in our case the apothem, which is given wrongly in the problem that has value 1.4m,and completely unnecessary.

Then proceed to calculate the area of one of the six equilateral triangles ,then the area of the hexagon, and finally the volume of the prism. The volume is finally approximately 2.92284 m³