The lateral edges of a regular hexagonal pyramid are all 20 cm long, and the base edges are all 16 cm long.

Question

To the nearest cc, what is the volume of this pyramid? To the mearest square cm, what is the combined area of the base and six lateral faces?

Answer 1

The volume of a pyramid is

 

V = (1/3)*base area*height

 

Since the base is a hexagon we use the formula for the area of a hexagon

 

base area = (3/2)√3*s²,  where s is the length of a side

 

base area =(3/2)√3*16² = 384√3

 

The height of the pyramid can be found using Pythagorean's Theorem

 

h = sqrt(20^2 - 16^2) = 12

 

The volume is

 

V = (1/3)(384√3)(12) = 1536√3 ≈ 2660 cm³

 

The find the surface area add up all the areas of the faces.

 

The area of the six lateral faces is 6 times the area of one.  The area of a triangle is

 

A = (1/2)*base*height

 

We have an isosceles triangle with sides 20 and base 10.  Dropping the perpendicular from to the base we bisect the base and can use Pythagorean's Theorem to find the height.

 

height = sqrt(20^2 - 5^2) = sqrt(375)

 

area of six lateral faces = 6*(1/2)*16*sqrt(375) = 48sqrt(375)

 

surface of pyramid = 384√3 + 48√375 ≈ 1595 cm²