Question

Find the lateral area, the total area, and the volume of this special figure.

Answer 1

Answer:

The $lateral\ surface\ area=72\sqrt{3} \ cm^{2}$, $total\ surface\ area=207.84\ cm^{2}$, and $volume=216\ cm^{3}$

Step-by-step explanation:

The question is to find the lateral surface area, The total area, and the volume of the hexagonal prism.

Lateral Surface Area (LSA)

      The lateral area does not include the top or bottom, only the side faces. One face of the hexagonal is a rectangle. There are six faces.

The area of the rectangle  is given as, $A=l\times b$

Where 'l' is length, here the height of the hexagonal is the length of the rectangle. That is, $l=3\sqrt{3}$

'b' is the breadth of the rectangle. That is, b=4

∴ $A=4\times 3\sqrt{3} =12\sqrt{3} \ cm^{2}$

So the lateral surface area of the hexagonal is six times the area of the rectangle. That is,

    $LSA=6\times 12\sqrt{3} \\\\LSA=72\sqrt{3} \ cm^{2}$.

Total Suraface Area (TSA)

  TSA=LSA+A₁+A₂

Where A₁ and A₂ are the areas of the top and bottom. The top and bottom are six-sided figures. The formula to find the area of the top or bottom is given as,

  $A=\dfrac{3a^2\sqrt{3}}{2}$

a=side of hexagon=4

   

          $\begin{array}{l}A=\dfrac{3\times4^2\times\sqrt{3}}{2}\\\\\ \ \ \ =24\sqrt{3}cm^{2}\end{array}$

Hence,

         $TSA=72\sqrt{3}+24\sqrt{3}+ 24\sqrt{3}\\\\TSA =207.84\ cm^{2}$

Volume (V)

The formula to calculate the volume of hexagonal given as,

      V=Ah

Where 'A' is area of the top or bottom and 'h' is the height of the hexagonal.

Hence,

          $V=24\sqrt{3}\times3\sqrt{3}\\\\V=216\ cm^{3}$