Ratio of Sides for Equal Area Shapes

We need to determine the ratio between the side length of a regular triangle and a regular hexagon when their areas are identical.

Key Formulas

• The area ($A_t$) of a regular triangle with side length '$s_t$' is given by:

  $ A_t = \frac{\sqrt{3}}{4} s_t^2 $

• The area ($A_h$) of a regular hexagon with side length '$s_h$' is given by:

  $ A_h = \frac{3\sqrt{3}}{2} s_h^2 $

Calculating the Side Ratio

The problem states that the areas are equal ($A_t = A_h$). Let's set the formulas equal to each other:

$ \frac{\sqrt{3}}{4} s_t^2 = \frac{3\sqrt{3}}{2} s_h^2 $

Now, we solve for the ratio $\frac{s_t}{s_h}$:

1. Cancel out $\sqrt{3}$ from both sides:

   $ \frac{1}{4} s_t^2 = \frac{3}{2} s_h^2 $

2. Multiply both sides by 4 to isolate $s_t^2$:

   $ s_t^2 = 4 \times \frac{3}{2} s_h^2 $

   $ s_t^2 = 6 s_h^2 $

3. Divide both sides by $s_h^2$ to get the ratio of the squares of the sides:

   $ \frac{s_t^2}{s_h^2} = 6 $

4. Take the square root of both sides to find the ratio of the sides:

   $ \frac{s_t}{s_h} = \sqrt{6} $

Therefore, the ratio of the side of the regular triangle to the side of the regular hexagon is $\sqrt{6}:1$.