Chapter 12.2, Problem 15CE

To determine

To find: The total area of the pyramid $V-XYZ$ .

Answer to Problem 15CE

The total area of the pyramid $V-XYZ$ is equal to $36\sqrt{3}$ .

Explanation of Solution

Given information:

Each edge of the pyramid $V-XYZ$ is equal to $6\text{cm}$ .

  

Calculation:

The total area of the pyramid is the sum of lateral area and area of base and the area of each face is equal to the area of the base.

  $\text{T}\text{.A}\text{.}=\left(n+1\right)\left(\text{B}\right)$

The base of the pyramid is equilateral triangle with edge of triangle is equal to $6\text{cm}$ .

The formula for the area of the equilateral triangle is,

  $\text{B}=\frac{\sqrt{3}}{4}{a}^2$ .

Substitute $6$ for $a$ in the formula for area.

  $\begin{array}{c}\text{B}=\frac{\sqrt{3}}{4}{a}^2\\ =\frac{\sqrt{3}}{4}{\left(6\right)}^2\\ =\frac{\sqrt{3}}{4}\left(36\right)\\ =9\sqrt{3}\end{array}$

Substitute $9\sqrt{3}$ for $\text{B}$ and $3$ for $n$ in the formula for lateral area.

  $\begin{array}{c}\text{T}\text{.A}\text{.}=\left(n+1\right)\left(\text{B}\right)\\ =4\left(9\sqrt{3}\right)\\ =36\sqrt{3}\end{array}$

Therefore, the total area of the pyramid $V-XYZ$ is equal to $36\sqrt{3}$ .