Question

A right pyramid is on a regular hexagonal base. Each side of the base is $10m.$ Its height is$60m.$ The volume of pyramid is
(A) $5196\,{m^3}$
(B) $5200\,{m^3}$
(C) $5210\,{m^3}$
(D) $51220\,{m^3}$

Answer 1

Hint: The regular hexagon is formed by six equilateral triangles. Use this concept to find the base area of the hexagon. And then, use the base area to calculate the volume of the pyramid by using the formula of volume of the pyramid in terms of base area.

Complete Step By Step Solution:
According to question the figure can be drawn as,

It is given in the question that the base of the pyramid is a regular hexagon. That means, all the sides of the hexagonal base will be the same.
Let $a = $ side of pyramid
It is given that each side of the base is $10\,m.$ means
$a = 10\, m.$
It is also given that, the height of the pyramid is $60\,m$
Let $h$ be the height of the pyramid. Then,
$h = 60\,m$
We have to find the volume of the pyramid.
Now, we have the formula for the volume of the pyramid as,
Volume of a pyramid$ = \dfrac{1}{3} \times $base area$ \times $height . . . (1)
To substitute the values in the above formula, first we have to find the Base area because it is needed in volume formula.
Now, since the base is a regular hexagon, the central angle of the hexagonal will be divided into 6 parts. Thus, the central angle of each section of the hexagon will be $ = \dfrac{{360}}{6} = {60^{\circ}}$
Since, it is a regular hexagon, we can conclude that, each part will be the equilateral triangle and all the angles will be ${60^{\circ}}$
Therefore, the base area of the regular hexagon will be equal to six times the area of one equilateral triangle.
Therefore, Base area$ = 6 \times \dfrac{{\sqrt 3 }}{4}{(a)^2}$
We know that $a = 10\,m$
Therefore, Base area $ = \dfrac{{3\sqrt 3 }}{2}{(10)^2}$
$ = \dfrac{{3\sqrt 3 }}{2} \times 100\,{m^2}$
Now put the value of Base area in volume formula, in equation (1).
Thus, Volume of the pyramid $ = \dfrac{1}{3} \times $base area$ \times $height.
$ = \dfrac{1}{3} \times \dfrac{{3\sqrt 3 }}{2} \times 100\,{m^2} \times 60\,m$
$ = 3\sqrt 3 \times 1000\,{m^3}$
Thus, the Volume of pyramid $ = 5196\,{m^3}$

Therefore, from the above explanation, the correct answer is, option (A) $5196{m^3}$

Note:
Knowing the properties of regular hexagons was critical in solving this question. You cannot calculate the base area of the hexagon until you come to the conclusion that the hexagon is formed by the six equilateral triangles. Once, you understand this, then the rest of the question was just substituting the values in the formula, which you need to know as well.