Chapter 12.7, Problem 18PPS

To determine

To find:Lateral and total surface area of each figure

Answer to Problem 18PPS

The lateral area of the square pyramid is $80c{m}_{}^2$

The surface area of the square pyramid is $144c{m}_{}^2$

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:

Consider the figure as shown below

  

The given figure is a square pyramid.

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

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

Here, $p$ is the perimeter of the base and $l$ is the slant height the square pyramid.

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

Here, $L$ is the lateral surface area and $B$ is the area of the base.

Find the lateral area of the square pyramid as follows:

The side of the square is $8cm.$

The perimeter $p$ of the square is $P=4\left(side\right)$ .

  $\begin{array}{l}P=4.8\\ =32\end{array}$

The slant of the square pyramid is $5cm.$

Substitute $P=32$ and $l=5$ in $L=\frac{1}{2}Pl$ .

  $\begin{array}{l}L=\frac{1}{2}Pl\\ =\frac{1}{\cancel{2}}.3{\cancel{2}}_{}^{16}.5\\ =16.5\\ =80\end{array}$

Hence, the lateral area of the square pyramid is $80c{m}_{}^2$

Find the surface area of the square pyramid as follows:

The area $B$ of the base is $B={\left(side\right)}_{}^2$

Thus,

  $\begin{array}{l}B={\left(8\right)}_{}^2\\ =64\end{array}$

Substitute $B=64$ and $L=180$ in $S=L+B$ .

  $S=L+B$

  $\begin{array}{l}=80+64\\ =144\end{array}$

Hence, the surface area of the square pyramid is $144c{m}_{}^2$

Conclusion:

The lateral area of the square pyramid is $80c{m}_{}^2$

The surfacearea of the square pyramid is $144c{m}_{}^2$