Chapter 12.7, Problem 6PPS

To determine

To find: the lateral and surface area

Answer to Problem 6PPS

The surface area of the pentagonal pyramid is $560{\text{yd}}^2$ and the lateral area of the pentagonal pyramid is $300{\text{yd}}^2$ .

Explanation of Solution

Given:

  

Formula used:

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

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

Calculation:

The given figure is a pentagonal pyramid.

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

Find the lateral area of the pentagonal pyramid as follows:

The side of the square is 10 yd.

The perimeter P of the pentagon is P = 5 (side)

Substitute side = 10 in the perimeter formula to find the perimeter of the pentagon.

Thus,

  $\begin{array}{c}P=5\cdot 10\\ =50\end{array}$

The slant height of the pentagonal pyramid is 12 yd.

Substitute P = 50 and l = 12 in the lateral surface area to find the lateral surface area of the pentagonal pyramid.

  $\begin{array}{c}L=\frac{1}{2}Pl\\ =\frac{1}{\cancel{2}}\cdot 50\cdot \cancel{1}{\cancel{2}}^6\\ =50\cdot 6\\ =300\end{array}$

Therefore, the lateral area of the pentagonal pyramid is $300{\text{yd}}^2$ .

Find the surface area of the pentagonal pyramid as follows:

The area B of the base is $260{\text{yd}}^2$

Substitute B = 260 and L = 300 in the surface area formula to find the surface area of the pentagonal pyramid.

  $\begin{array}{c}S=L+B\\ =300+260\\ =560\end{array}$

Conclusion:

Therefore, the surface area of the pentagonal pyramid is $560{\text{yd}}^2$ and the lateral area of the pentagonal pyramid is $300{\text{yd}}^2$ .