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How do you calculate the surface area of the lateral faces of a regular hexagonal pyramid that has a slant height of 35 cm and a base side length of 4 cm?

Answer
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Hint: We recall the shape of a regular hexagonal pyramid which has lateral faces in congruent triangular shape. We find the area of one triangular base and multiply 6 to get the answer. We find the area of one triangle as 12×base×height=12×a×l where a is the base side length and l is the slant height.

Complete step by step answer:
We know that a Pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle, called a lateral face. A right pyramid has its apex directly above the centroid of its base. A pyramid that has a regular polygon base is called a regular pyramid. A regular hexagonal pyramid has its base as a regular hexagon. We draw the rough figure of it below with O as the apex and ABCDEF as the base of a regular hexagonal base.
 
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The altitude dropped from apex on the hexagon is called height of the pyramid h=OG and the base side length here is a=AB=BC=CD=DE=EF=FA. Here l is the slant height dropped from apex on one of the sides (Here OH).
 The surfaces excluding the base are called lateral surfaces and in a regular pyramid they are congruent. We have 6 lateral faces in a regular hexagonal pyramid which are in triangular shape. Since they are congruent they will be of equal area. So the surface area of the lateral faces will be 6 times the area of one triangular face.
Let us observe the triangle OEH. We know that the area of one triangle is 12×base×height. Here base is EF=a and height of triangle is slant height l=OH. So area of triangle OEH is
A=12×a×l
 The regular hexagonal pyramid has a slant height of 35 cm and a base side length of 4 cm. So we have l=35 cm, a=4cm. So area of one triangle is
A=12×4×35
We multiply 6 and find the surface area of all the lateral faces (LSA) as
L.S.A=6A=6×12×4×35=3×4×35=420 cm2

Note: We note that the altitude from the centre of the hexagon on a side (hereGH=p) is called apothem. We can find the total surface area of the hexagonal pyramid as 3pa+3la where 3pa is the base area and 3la is the sum of lateral surface area. We can find the volume of the hexagonal pyramid as V=p×a×h.