The base of the right pyramid is a regular hexagon whose area is 243 square cm. if the area of a side face of the pyramid is 46 square cm, then its volume (in cc.) in the nearest will be ____.

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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.
The volume of pyramid is given with b as the area of the base and h as the height of from the base to the apex also is given as

V = 13 b h

.
We are given the question that the base of a right pyramid is a regular hexagon whose area is 243 square cm which means b = 243 square cm. The area of a side face of the pyramid is 46 square cm. We draw the rough figure of it below with O as the apex and ABCDEF as the base of a regular hexagonal base.

https://www.vedantu.com/question-sets/e521b027-f0fe-483e-945c-c56b7b22b7fa6684921080437784688.png

Here l is the slant height dropped from apex on one of the sides (Here OH). The line segment GH is called apothem whose length we denote as pp. We know that area of regular hexagon with side a is 6 times the area of equilateral triangle with side aa which means

V = 6 × 34 a 2

. So we have;

6 × 34 a 2 = 243 ⇒ a 2 = 2436 × 43 ⇒ a 2 = 1623 = 93.53 ⇒ a = 9.67

We observe the triangle OEF which is a die face of the pyramid with slant height

l = O H

and base

a = E F = 9.67 cm

. We are given its area as

46 s q u a r e cm

. So we have;

12 × l × a = 46 ⇒ l = 46 × 2 a ⇒ l = 92 9.67 = 9.81

We can find the apothem

p

as the height of equilateral triangle EGF as

p = 32 a = 32 × 9.67 = 8.37

We use Pythagoras theorem in right angled triangle OGH to have;

h = l 2 − p 2 ⇒ h = 9.81 2 − 8.37 2 ⇒ h = 5.12

So the volume of pyramid cubic

cm

is

V = 13 b h = 13 × 243 × 5.12 = 414.44

The base of the right pyramid is a regular hexagon whose area is 243 square cm. if the area of a side face of the pyramid is 46 square cm, then its volume (in cc.) in the nearest will be ____.

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