Step-by-step Answer:
The given pyramid has a hexagonal (6-sides) base, which has the property that it is made up of six equilateral triangles each with side 8cm (same as the external side).
The apothem of each equilateral triangle is equal to sqrt(3)/2 of the side, namely 8sqrt(3)/2 = 4sqrt(3).
So area of the base is six times that of each equilateral triangle, whose individual area is given by A=base*height/2. So
area of equilateral triangle = (8) * 4sqrt(3) / 2 = 27.713
Area of base (6 equilateral triangles = 6*27.713
= 166.277 cm^2
To find the area of each slanted face, we need to calculate the apothem of each face, namely by using Pythagoras (see attached figure, on top left).
Apothem = sqrt(11^2+(4*sqrt(3))^2) = sqrt(121+48) = sqrt(169) = 13
Area of each slanted face = base * height (apothem) /2 = 8*13/2= 52
Area of six slanted faces = 6 * 52 = 312 cm^2
Total surface area = area of base + area of slanted faces
= 166.277 + 312
= 478.28 cm^2 (to hundredth cm^2)
