Two regular hexagons have their sides in the ratio 2:3.Find the ratio of the areas of the 1 St & 2nd hexagons?
area grows as the square of the sides.
So, the areas are in the ratio (2/3)^2 = 4/9
areas of similar shapes are proportional to the square of their corresponding sides
ratio of their areas is 2^2 : 3^2 = 4 : 9
To find the ratio of the areas of the first and second hexagons, we need to consider that the ratio of their sides is given as 2:3.
The formula to calculate the area of a regular hexagon is A = (3√3/2) * s², where s is the length of each side.
Let's assume the length of the sides of the first hexagon is 2x (based on a ratio of 2:3). Thus, the length of the sides of the second hexagon would be 3x.
Now we can calculate the areas of the hexagons:
Area of First Hexagon, A₁ = (3√3/2) * (2x)² = (3√3/2) * 4x² = (12√3) * x²
Area of Second Hexagon, A₂ = (3√3/2) * (3x)² = (3√3/2) * 9x² = (27√3) * x²
Therefore, the ratio of the areas of the first and second hexagons is:
A₁:A₂ = (12√3) * x² : (27√3) * x²
Simplifying the expression, we can cancel out the common factors:
A₁:A₂ = (12√3)/(27√3) * x²/x²
= 4/9
Hence, the ratio of the areas of the first and second hexagons is 4:9.
To find the ratio of the areas of the two hexagons, we need to know the relationship between their side lengths and their areas.
The area of a regular hexagon can be found using the formula: A = (3√3 / 2) * s^2, where s is the length of each side.
Let's assume that the first hexagon has side length 2x, and the second hexagon has side length 3x.
Now we can calculate the areas of the hexagons:
For the first hexagon:
Area of the 1st hexagon = (3√3 / 2) * (2x)^2 = (3√3 / 2) * 4x^2 = 6√3x^2
For the second hexagon:
Area of the 2nd hexagon = (3√3 / 2) * (3x)^2 = (3√3 / 2) * 9x^2 = 27√3x^2
Now, let's find the ratio of their areas:
Ratio = (Area of the 1st hexagon) / (Area of the 2nd hexagon)
= (6√3x^2) / (27√3x^2)
= 6√3x^2 divided by 27√3x^2
= (6/27) * (√3x^2 / √3x^2)
= (2/9) * 1
= 2/9
Therefore, the ratio of the areas of the first and second hexagons is 2:9.