Chapter 12.7, Problem 7PPS

To determine

To find: the lateral and surface area

Answer to Problem 7PPS

The surface area of the triangular pyramid is $140.4{\text{mm}}^2$ and the lateral area of the triangular pyramid is $105.3{\text{mm}}^2$

Explanation of Solution

Given:

Consider the figure as shown below:

  

Formula used:

The lateral area L of a triangularpyramid is given by the formula $L=\frac{1}{2}Pl$

The surface area S of a triangularpyramid is given by the formula S = L + B

Calculation:

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

First find the lateral area of the triangularpyramid.

The side of the triangle is 9 mm, 9 mm, and 9 mm.

The perimeter P of the triangle is

  $\begin{array}{c}P=\text{sum}\text{of}\text{sides}\\ =9+9+9\\ =27\end{array}$

The slant height of the square pyramid is 7.8 mm.

Put P = 27 and l = 7.8 in the lateral surface area to find the lateral surface area of the square pyramid.

  $\begin{array}{c}L=\frac{1}{2}Pl\\ =\frac{1}{\cancel{2}}\cdot 27\left(\cancel{7}.{\cancel{8}}^{3.9}\right)\\ =27\left(3.9\right)\\ =105.3\end{array}$

Therefore, the lateral area of the triangular pyramid is $105.3{\text{mm}}^2$

Next find the surface area of the square pyramid.

First find the area B of the triangular pyramid. The area B of the base is given by $B=\frac{1}{2}\cdot b\cdot a$ where b is the base and a is the altitude of the triangle

  $\begin{array}{c}=\frac{1}{\cancel{2}}\cdot 9\left(\cancel{7}.{\cancel{8}}^{3.9}\right)\\ =9\left(3.9\right)\\ =35.1\end{array}$

Put B = 35.1 and L = 105.3 in the surface area formula to find the surface area of the triangular pyramid.

  $\begin{array}{c}S=L+B\\ =105.3+35.1\\ =140.4\end{array}$

Conclusion:

Therefore, the surface area of the triangular pyramid is $140.4{\text{mm}}^2$ and the lateral area of the triangular pyramid is $105.3{\text{mm}}^2$