Question

Find the volume of a regular hexagonal based prism with base side 4 cm and height 10 cm.

Answer 1

Hint: In this question, as the base of the prism is a hexagon we need to find the area of this base using the formula of area of a hexagon given by the formula \[\dfrac{3\sqrt{3}}{2}{{a}^{2}}\]. Now, on multiplying this area of the base which is calculated with the given height of the prism in the question given by the formula \[\dfrac{3\sqrt{3}}{2}{{a}^{2}}h\] gives the volume of the prism with base as hexagon.

Complete step-by-step solution -
RIGHT PRISM:
A right prism is a prism in which the joining edges and faces are perpendicular to the base faces
\[\text{volume}=\text{area of base}\times \text{height}\]
Here, given in the question that the base of the prism is a hexagon
Now, as we already know that the area of a hexagon is given by the formula
\[\dfrac{3\sqrt{3}}{2}{{a}^{2}}\]
Now, from the given condition in the question about the side of the base we have
\[a=4\]
Now, on substituting the respective value of side in the area of a hexagon we get,
\[\Rightarrow \dfrac{3\sqrt{3}}{2}\times {{4}^{2}}\]
Now, on further simplification we get,
\[\Rightarrow 24\sqrt{3}c{{m}^{2}}\]
As we already know that the formula for volume of a prism is given by
\[\Rightarrow \text{volume}=\text{area of base}\times \text{height}\]
Here, as the base here is a hexagon we need to substitute the area of a hexagon
\[\Rightarrow \dfrac{3\sqrt{3}}{2}{{a}^{2}}h\]
As already given in the question that the height of the prism as
\[h=10\]

Now, on substituting the respective values in the above formula of volume of the prism we get,
\[\Rightarrow \dfrac{3\sqrt{3}}{2}\times {{4}^{2}}\times 10\]
Now, this can be further written in the simplified form as
\[\Rightarrow 24\sqrt{3}\times 10\]
Now, on further simplification we get,
\[\Rightarrow 240\sqrt{3}c{{m}^{3}}\]
Hence, the volume of the given prism with hexagon base is \[240\sqrt{3}c{{m}^{3}}\].

Note: Instead of finding the area of the hexagon separately we can directly consider the volume formula including the area formula and then substituting the respective value of side and simplify further. This method reduces the number of steps.It is important to note that the area of the hexagon can also be found by multiplying the area of equilateral triangle of given side with 6. While calculating the volume and area we need to substitute the respective values of height and side because substituting in the other way changes the result completely.