Chapter 11.5, Problem 16PSB

To determine

To Calculate: Ratio of the area of the smaller hexagon to the area of the larger hexagon.

Answer to Problem 16PSB

The ratio of the area of the smaller hexagon to the area of the larger hexagon is $3:4$

Explanation of Solution

Given information:

Radius of the circle is $6$

Formula used:

Here, we are using the formula

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

Calculation:

Here, Area of smaller hexagon = $6\times \frac{\sqrt{3}}{4}\times 6\times 6$

  → $54\sqrt{3}$

Now, Area of larger circle = $2\sqrt{3}{r}^2$

  → $2\sqrt{3}\times 36$

  → $72\sqrt{3}$

So, $\frac{ar\left(smaller\right)}{ar\left(l\arg er\right)}=\frac{54\sqrt{3}}{72\sqrt{3}}$

  → $\frac{3}{4}$

Conclusion: Hence,the ratio of the area of the smaller hexagon to the area of the larger hexagon is $3:4$