Chapter 11.7, Problem 7PSA

To determine

To find: the ratio of the areas of two regular hexagons with sides $2$ and $8$ respectively.

Answer to Problem 7PSA

The ratio of the areas of two regular hexagons is $1:16.$

Explanation of Solution

Given Information:

Side of 1^st hexagon $\left(a_1\right)=2$

Side of 2^nd hexagon $\left(a_2\right)=8$

Formula used:

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

Here, a is the side of a regular hexagon

Calculation:

We know that, Area of a regular hexagon $=\frac{3\sqrt{3}}{2}{a}^2$ … (i)

In the 1^st regular hexagon, side ( a₁ ) = 2.

In (i), take $a=a_1=2$

Area of the 1^st hexagon $=\frac{3\sqrt{3}}{2}\times 2^2=3\sqrt{3}\times 2=6\sqrt{3}$

In the second regular hexagon, side ( a₂ ) = 8.

In (i), take $a=a_2=2$

Area of the 2^nd hexagon $=\frac{3\sqrt{3}}{2}\times 8^2=3\sqrt{3}\times 32=96\sqrt{3}$

Ratio of areas of two hexagons $=\frac{Areaof{1}^{st}hexagon}{Areaof{2}^{nd}hexagon}=\frac{6\sqrt{3}}{96\sqrt{3}}=\frac{1}{16}$

Hence, the ratio of areas of the regular hexagons is $1:16.$