Chapter 12, Problem 6CR

To determine

To find:The slant height and lateral area of a regular pentagonal pyramid.

Answer to Problem 6CR

Theslant height and lateral area are $4$ and $60$ respectively.

Explanation of Solution

Given information:

A regular pentagonal pyramid has base edge 6 and lateral edge 5.

Formula used: The formula used to find the lateral area of the pentagonal pyramid is:

  $\text{Lateral area}=\frac{1}{2}\times \text{perimeter}\times \text{slant height}$

Calculation:

Draw a figure of pentagonal pyramid. The height from the apex of pyramid to the center of the pyramid, and hypotenuse is known as the slant height of the pyramid. So, draw a right angle triangle to find its slant height.

  

The triangle $\Delta AOE$ is a right angle triangle. So, the slant height is calculated as:

  $\begin{array}{c}{x}^2+3^2=5^2\\ x^2=25-9\\ x=4\end{array}$

The perimeter of the base of pentagonal pyramid is the sum of all sides.

  $\begin{array}{c}\text{perimeter}=6+6+6+6+6\\ =30\end{array}$

Now, the lateral area can be calculated as:

  $\begin{array}{c}\text{Lateral area}=\frac{1}{2}\times \text{perimeter}\times \text{slant height}\\ \text{=}\frac{1}{2}\times 30\times 4\\ =60\end{array}$

Therefore, the slant height and lateral area are 4 and 60 respectively.