Chapter 14.6, Problem 23PSD

To determine

To construct theequilateral triangle whose area is equal to that of hexagon.

Explanation of Solution

Given:

The given regular hexagon.

Calculation:

The equilateral triangle whose area is equal to that of hexagoncan be constructed as follows-

Step 1- First construct the regular hexagon.

Step 2- Let the area of regular hexagon be A.

Step 3- Now, it is know that the area of equilateral triangle with side $a$ is $\frac{\sqrt{3}}{4}{a}^2$

Step 4- These areas can be equal when

  $\begin{array}{l}\frac{\sqrt{3}}{4}{a}^2=A\\ a^2=\frac{4a}{\sqrt{3}}\end{array}$

Step 5- construct the equilateral triangle with the side being $a^2=\frac{4a}{\sqrt{3}}$

Step 6- Draw the angle ${60}^{\circ }$ at base.

Step 7- Join all the vertices to get the required equilateral triangle.