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 $\frac{s_t}{s_h}$:
- Cancel out $\sqrt{3}$ from both sides:
$ \frac{1}{4} s_t^2 = \frac{3}{2} s_h^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 $
- 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 $
- 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 $\sqrt{6}:1$.
Similar Questions
If the perimeter of a regular octagon is 80 cm, what is the side length?
The ratio of areas of two regular polygons with same number of sides is 4:9. What is the ratio of their side lengths?
A regular polygon has each interior angle measuring 150°. Find the number of its sides.
Important Questions from Plane Figures
If the area of a square is 625 cm 2, then what is the perimeter of the square?
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