Chapter 12.7, Problem 8PPS

To determine

To find: the lateral and surface area of the triangular pyramid.

Answer to Problem 8PPS

The surface area of the triangular pyramid is $87.6{\text{in}}^2$ andthe lateral area of the triangular pyramid is $72{\text{in}}^2$

Explanation of Solution

Given:

Base side length is 6 in.

Base area is $15\frac{3}{5}{\text{in}}^2$

Slant height is 8 in.

Formula used:

The lateral area L of a triangular pyramid is given by the formula

  $L=\frac{1}{2}Pl$

The surface area S of a triangular pyramid is given by the formula

S = L + B

Calculation:

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

First convert the mixed number into decimal.

  $\begin{array}{c}15\frac{3}{5}=\frac{15\cdot 5+3}{5}\\ =\frac{75+3}{5}\\ =\frac{78}{5}\\ =15.6\end{array}$

First find the lateral area of the triangular pyramid.

The perimeter P of the triangular is

P = 3(side)

Put side = 6 in the perimeter formula to find the perimeter of the triangular Thus,

  $\begin{array}{c}P=3\cdot 6\\ =18\end{array}$

The slant height of the triangular pyramid is 8 in.

Put P = 18 and l = 8 in the lateral surface area to find the lateral surface area of the triangular pyramid.

  $\begin{array}{c}L=\frac{1}{2}Pl\\ =\frac{1}{\cancel{2}}\cdot 18\cdot {\cancel{8}}^4\\ =18\cdot 4\\ =72\end{array}$

Therefore, the lateral area of the triangular pyramid is $72{\text{in}}^2$

Next find the surface area of the triangular pyramid

The area B of the base is $15.6{\text{in}}^2$

Put B = 15.6 and L = 72 in the surface area formula to find the surface area of the triangular pyramid.

  $\begin{array}{c}S=L+B\\ =72+15.6\\ =87.6{\text{in}}^2\end{array}$

Conclusion:

Therefore, the surface area of the triangular pyramid is $87.6{\text{in}}^2$ andthe lateral area of the triangular pyramid is $72{\text{in}}^2$