Side of a hexagon is 4cm . Side of a square is `4sqrt(2)` cm . What is the ratio of their areas ?

Options

A. `3sqrt(3) : 2`
B. `3sqrt(3) : 4`
C. `sqrt(3)//2 : 1`
D. `2sqrt(3) : 1`

Answer

To find the ratio of the areas of a hexagon and a square, we will follow these steps:

### Step 1: Calculate the area of the hexagon
The formula for the area of a regular hexagon with side length \( a \) is given by:

\[ \text{Area} = \frac{3\sqrt{3}}{2} a^2 \]

Given that the side of the hexagon is \( 4 \, \text{cm} \):

\[ \text{Area of hexagon} = \frac{3\sqrt{3}}{2} (4)^2 = \frac{3\sqrt{3}}{2} \times 16 = 24\sqrt{3} \, \text{cm}^2 \]

### Step 2: Calculate the area of the square
The formula for the area of a square with side length \( a \) is:

\[ \text{Area} = a^2 \]

Given that the side of the square is \( 4\sqrt{2} \, \text{cm} \):

\[ \text{Area of square} = (4\sqrt{2})^2 = 16 \times 2 = 32 \, \text{cm}^2 \]

### Step 3: Find the ratio of the areas
Now, we need to find the ratio of the area of the hexagon to the area of the square:

\[ \text{Ratio} = \frac{\text{Area of hexagon}}{\text{Area of square}} = \frac{24\sqrt{3}}{32} \]

### Step 4: Simplify the ratio
To simplify the ratio, we can divide both the numerator and the denominator by 8:

\[ \frac{24\sqrt{3}}{32} = \frac{3\sqrt{3}}{4} \]

### Final Answer
Thus, the ratio of the areas of the hexagon to the square is:

\[ \text{Ratio} = 3\sqrt{3} : 4 \]

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