Chapter 11, Problem 5PT

To determine

To find: The area of each cube in the given pattern.

Answer to Problem 5PT

The area of each cube in the pattern is $9{\text{ in}}^{\text{2}}$ .

Explanation of Solution

Given information:

The length of the diagonals of each Rhombus in the pattern $d_1=2\text{ and }{d}_2=3$.

Formula used:

Area of the rhombus, $A=\frac{1}{2}{d}_1{d}_2$ ; where $d_1\text{ and }{d}_2$ are the length of the diagonals.

Calculation:

Calculate the area of a rhombus in the cube as follows:

  $\begin{array}{c}{A}_{\text{rhombus}}=\frac{1}{2}\left(2\right)\left(3\right)\\ =\frac{6}{2}\\ =3\end{array}$

Thus, the area of each rhombus is $A=3{\text{ in}}^{\text{2}}$ .

From the pattern it is observed that each cube is made up of three rhombus.

So, the area of each cube is calculated as follows:

  $\begin{array}{c}{A}_{\text{cube}}=3\times A_{\text{rhombus}}\\ =3\times 3\\ =9\end{array}$

Therefore, the area of each cube in the pattern is $9{\text{ in}}^{\text{2}}$ .