Answer

Verified

446.4k+ views

**Hint:**First of all, draw the diagram of the equilateral triangle-based prism to have a clear idea of what we have to find. Then use the formula for finding L.S.A and T.S.A of the equilateral triangle-based prism.

**Complete step-by-step solution:**Given that the base of the prism is an equilateral triangle with side 4 cm as shown in the given figure:

Let \[p\] be the perimeter of the base of the prism i.e., the perimeter of equilateral \[\Delta ABC\].

So, \[p = 4 + 4 + 4 = 12{\text{ cm}}\]

Given height of the prism \[h = 8\;{\text{cm}}\]

We know that the lateral surface area of the triangle-based prism is given by \[p \times h\] where \[p\] is the perimeter of the base triangle and \[h\] is the height of the prism.

Therefore, L.S.A of the prism \[ = p \times h = 12 \times 8 = 96{\text{ c}}{{\text{m}}^2}\]

Now, consider the area of equilateral \[\Delta ABC\].

We know that the area of the equilateral triangle with side \[a\] is given by \[\dfrac{{{a^2}}}{4}\].

So, area of \[\Delta ABC = \dfrac{{\sqrt 3 {{\left( 4 \right)}^2}}}{4} = 4\sqrt 3 \]

We know that the total surface area of the triangle based prism with base area \[A\], perimeter \[p\] and \[h\] is the height of the prism is given by \[2 \times A + p \times h\].

Therefore, the total surface area of the prism \[ = 2 \times 4\sqrt 3 + 12 \times 8\]

\[

= 8\sqrt 3 + 96 \\

= 8\left( {1.73} \right) + 96 \\

= 13.84 + 96 \\

= 109.84{\text{ c}}{{\text{m}}^2} \]

**Thus, the L.S.A of the prism is \[96{\text{ c}}{{\text{m}}^2}\] and the T.S.A of the prism is \[{\text{109}}{\text{.84 c}}{{\text{m}}^2}\].**

**Note:**The lateral surface area of the triangle-based prism is given by \[p \times h\] where \[p\] is the perimeter of the base triangle and \[h\] is the height of the prism. The total surface area of the triangle based prism with base area \[A\], perimeter \[p\] and \[h\] is the height of the prism is given by \[2 \times A + p \times h\].

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Mark and label the given geoinformation on the outline class 11 social science CBSE

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE