In solid geometry, the volume of any pointed solid is equal to one-third of the product of the area of its base and the vertical height. In equation, it is written as:
V = (1/3)*Bh
where
B is area of the base
h is height
We know the height to be 27 in. The missing information is the area of the hexagonal base. So, let's focus on this shape as drawn in the attached picture. The area of any given polygon is one-half the product of its apothem and perimeter:
B = (1/2)*aP
The apothem is a line drawn from the center of the polygon and projected down to the center of its base. So, an apothem is a perpendicular bisector denoted by the red line. Each interior angle of a hexagon is equal to 60° because one revolution divided by 6 sides is: 360/6 = 60°. When an apothem is drawn, it makes a right angle with an angle of 30° and a base of half of 23 inches. Using the pythagorean theorem, the apothem is equal to:
tan 30° = (23/2) ÷ a
a = 23*sqrt(3)/2 in
The perimeter is the sum of all sides of the polygon. Assuming the hexagon is regular, all its sides are equal measuring 23 inches. So, the perimeter is equal to:
P = 23(6) = 138 in
So, the area of the base is equal to:
B = (1/2)(23*sqrt(3)/2 in)(138 in)
B = 1,374.3823 in²
So, we can finally solve for V:
V = (1/3)(1,374.3823 in²)(27 in)
V = 12,369.44 in³
