Chapter 12, Problem 12CT

To determine

To find:The area of the smaller pyramid with height 6.

Answer to Problem 12CT

The area of the smaller pyramid is $72$ .

Explanation of Solution

Given information:The height of a regular pyramid is 18 and total area is 648. The height of a similar pyramid is 6.

Calculation:

The ratio of the corresponding sides of any two similar geometric figures is termed as Scale factor. The theorem of similar solids states that if the scale factor of two similar solids is $a:b$ , then

(i) The ratio of base areas, lateral areas and total areas is $a^2:b^2$ .

(ii) The ratio of corresponding sidesis $a:b$ .

(iii) The ratio of their volumes is $a^3:b^3$ .

The ratio of corresponding heights gives the scale factor of similar pyramids as:

  $\begin{array}{c}\frac{a}{b}=\frac{18}{6}\\ =\frac{3}{1}\end{array}$

The ratio of total area of both pyramids is:

  $\begin{array}{c}{\left(\frac{a}{b}\right)}^2={\left(\frac{3}{1}\right)}^2\\ =\frac{9}{1}\end{array}$

Now, the total area of smaller pyramid can be calculated as:

  $\begin{array}{c}\frac{\text{total}\text{area}\text{of}\text{pyramid}}{\text{total}\text{area}\text{of}\text{smaller}\text{pyramid}}=\frac{9}{1}\\ \frac{648}{\text{total}\text{area}\text{of}\text{smaller}\text{pyramid}}=\frac{9}{1}\\ \text{total}\text{area}\text{of}\text{smaller}\text{pyramid}=\frac{648}{9}\\ =72\end{array}$

Therefore, the area of smaller pyramid is 72.