Chapter 11.5, Problem 17PSB

a.

To determine

To calculate: Area of the region between the square and triangle.

Answer to Problem 17PSB

The area of the shaded region is $36-9\sqrt{3}{\text{ units}}^2$

Explanation of Solution

Given: The side of the square ( a ) = $6$

  

Concept used: Here, we are using the formula

Area of triangle = $\frac{\sqrt{3}}{4}\times a^2$

Where. “ a ” is the side of the triangle

Area of square = $a^2$

Where, $"a"$ is side of the square

Calculation:The area of square is given as

  $\begin{array}{l}{A}_s=a^2\\ A_s=6^2\\ A_s=36\end{array}$

Now, The area of triangle

  $A_t=\frac{\sqrt{3}}{4}\times a^2$

  = $\frac{\sqrt{3}}{4}\times 6\times 6$

  = $9\sqrt{3}$

So, the area of shaded region

  $\begin{array}{l}=A_s-A_t\\ =36-9\sqrt{3}\end{array}$

Conclusion: Hence, the area of the required region $36-9\sqrt{3}{\text{ units}}^2$

b.

To determine

To calculate:Area of the region between the hexagon and triangle.

Answer to Problem 17PSB

The area of the shaded region is $27\sqrt{3}$

Explanation of Solution

Given: The side of the hexagon is $6$

  

Formula used:

Here, the formula is

Area of hexagon = $\frac{3\sqrt{3}}{2}\times a^2$

Calculation:

Area of hexagon = $\frac{3\sqrt{3}}{2}\times 36$

  → $54\sqrt{3}$

Area of triangle = $\frac{1}{2}\times 3\times 3\times 6\sqrt{3}$

  → $27\sqrt{3}$

Now, Area of shaded region = $54\sqrt{3}-27\sqrt{3}$

  → $27\sqrt{3}$

Conclusion: Hence, the area of the required region is $27\sqrt{3}$

c.

To determine

To calculate:Area of the region between the hexagon and rectangle.

Answer to Problem 17PSB

The area of the shaded region is $36\sqrt{3}$

Explanation of Solution

Given: The side of the hexagon is $6$

  

Concept used: Here, we are using the formula

Area of rectangle = $length\times breadth$

Calculation: Tan ${30}^{\circ }=\frac{P}{B}$ (here, $"P"$ is perpendicular and $"B"$ is base)

  $\frac{1}{\sqrt{3}}=\frac{3}{B}$

  $B=3\sqrt{3}$

So, length = $3\sqrt{3}+3\sqrt{3}=6\sqrt{3}$

And, breadth = $6$

So, Area of rectangle = $6\sqrt{3}\times 6$

Conclusion: Hence, the area of the shaded region is $36\sqrt{3}$