Chapter 12.7, Problem 3CYU

To determine

To find: the lateral and surface area

Answer to Problem 3CYU

The surface area of the square pyramid is $336{\text{m}}^2$ and the lateral area of the square pyramid is $192{\text{m}}^2$

Explanation of Solution

Given:

  

Formula used:

The lateral area L of a square pyramid is $L=\frac{1}{2}Pl$ .

The surface area S of a square pyramid is S = L + B

Calculation:

The given figure is a square pyramid.

The objective is to find the lateral and surface area of the square pyramid.

Find the lateral area of the square pyramid.

The side of the square is 12 mm.

The perimeter P of the square is P = 4(side)

Substitute side = 12 in the perimeter formula to find the perimeter of the square.

Thus,

  $\begin{array}{c}P=4\cdot 12\\ =48\end{array}$

The slant height of the square pyramid is 8 mm.

Substitute P = 48 and l = 8 in the lateral surface area to find the lateral surface area of the square pyramid.

  $\begin{array}{c}L=\frac{1}{2}Pl\\ =\frac{1}{\cancel{2}}\cdot 48\cdot {\cancel{8}}^4\\ =48\cdot 4\\ =192\end{array}$

Therefore, the lateral area of the square pyramid is $192{\text{m}}^2$

Find the surface area of the square pyramid.

The area B of the square pyramid is.

  $\begin{array}{c}B={\left(side\right)}^2\\ ={12}^2\\ =144\end{array}$

Substitute B = 144 and L = 192 in the surface area formula to find the surface area of the square pyramid.

  $\begin{array}{c}S=L+B\\ =192+144\\ =336\end{array}$

Conclusion:

Therefore, the surface area of the square pyramid is $336{\text{m}}^2$ and the lateral area of the square pyramid is $192{\text{m}}^2$