Ratio of Equilateral Triangle Area to Square Area

Area of Equilateral Triangle

The area of an equilateral triangle with side length $s$ is given by the formula:

Area_triangle = $\frac{\sqrt{3}}{4} s^2$

For a triangle with side $x$, the area is:

Area_triangle = $\frac{\sqrt{3}}{4} x^2$

Area of Square

The area of a square with side length $s$ is given by the formula:

Area_square = $s^2$

For a square with side $x$, the area is:

Area_square = $x^2$

Calculating the Ratio

The question asks for the ratio of the area of the equilateral triangle to the area of the square.

Ratio = Area_triangle : Area_square

Substitute the formulas:

Ratio = $\left( \frac{\sqrt{3}}{4} x^2 \right) : (x^2)$

To simplify the ratio, divide both sides by $x^2$ (assuming $x \neq 0$):

Ratio = $\frac{\sqrt{3}}{4} : 1$

Multiply both sides by 4 to eliminate the fraction:

Ratio = $\sqrt{3} : 4$

The final ratio is $\sqrt{3} : 4$.