Q17. - Geometry

Question

Facts about the base of a right prism and the height of the prism are given. Sketch each prism and find its lateral area, total area and volume.

Equilateral triangle with side 8, h = 10.

Step-by-Step Solution

Answer

The lateral area is 240, the total area is$(240+32\sqrt{3})$ and the volume is$160\sqrt{3}$ .

Step 1. Given information.

The side of equilateral triangle is 8 and height is 10.

Step 2. Write the concept.

The lateral area of the equilateral triangular prism is given by:

$\text{L.A.}=3(\text{Side}\times \text{Height})$

 The total area of the equilateral triangular prism is given by:

$\text{T.A.}=\text{L.A.+2}\left(\frac{\sqrt{3}}{4}{\text{Side}}^2\right)$

 The volume of equilateral triangular prism is given by:

$\text{V}=\frac{\sqrt{3}}{4}{\left(\text{side}\right)}^2\times h$

Step 3. Sketch the graph.

Now to draw a prism first make its base as a triangle and then sketch the length of 10 triangle from its each point, 

Thus, the prism can be obtained as:

Step 4. Determine values.

The lateral area is:

$\begin{array}{c}\text{L.A.}=3(\text{Side}\times \text{Height})\\ =3(8\times 10)\\ =3\times 80\\ =240{\text{\hspace{0.17em}\hspace{0.17em}unit}}^2\end{array}$

 

The total area is:

$\begin{array}{c}\text{T.A.}=\text{L.A.+2}\left(\frac{\sqrt{3}}{4}{\text{Side}}^2\right)\\ =240+\frac{\sqrt{3}}{2}{\left(8\right)}^2\\ =240+\frac{\sqrt{3}}{2}\times 64\\ =(240+32\sqrt{3}){\text{unit}}^2\end{array}$

 

And the volume is:

$\begin{array}{c}\text{V}=\frac{\sqrt{3}}{4}{\left(\text{side}\right)}^2\times h\\ =\frac{\sqrt{3}}{4}{\left(\text{8}\right)}^2\times 10\\ =\frac{64\sqrt{3}}{4}\times 10\end{array}$

 

So,

$\begin{array}{c}\text{V}=16\sqrt{3}\times 10\\ =160\sqrt{3}{\text{\hspace{0.17em}\hspace{0.17em}unit}}^3\end{array}$