Question

The diagonal of a square is twice the side of equilateral triangle the ratio of area of the triangle to the area of square is?

Answer 1

Let the length of the side of the equilateral triangle be x

The diagonal of the square will be 2x

Find the length of the square:

a² + b² = c²

Length² + Length² = (2x)²

2Length² = 4x²

Length² = 2x²

Length = (√2) x

Find the area of the square:

Area = Length²

Area = [  (√2) x ] ²

Area = 2x² units²

Find the area of the triangle:

Area = √3/4 (Side)²

Area = √3/4 (x)² units²

Find the ratio:

Area of the triangle : Area of square = √3/4 (x)²  : 2x²

Divide by x² through:

Area of the triangle : Area of square = √3/4  : 2

Divide by 4 through:

Area of the triangle : Area of square = √3  : 8

Answer: The ratio is √3  : 8

Answer 2

Let us consider one side of the equilateral triangle is 1 unit, area of the equilateral triangle is √3/4 a², then the area of an equilateral triangle for  1 unit is √3/4.

The diagonal of the square is two sides of equilateral triangle equal two units.

Area of the square is ½ D², D is diagonal.

The area of triangle = ½ (2²) = 2, triangle: square =  √3/4: 2, the ratio of the equilateral triangle and square is √3:8.