A kite in the shape of a square with a diagonal 32 cm attached to an equilateral triagle of the base 8 cm. Approximately how much paper has been used to make it ? (Use `sqrt(3) = 1.732`)

Options

A. `539.712 cm^(2)`
B. `538.721 cm^(2)`
C. `540.712 cm^(2)`
D. `539.217 cm^(2)`

Answer

To find the total area of the kite made of a square and an equilateral triangle, we will follow these steps:

### Step 1: Calculate the area of the square
The area of a square can be calculated using the formula:
\[ \text{Area of square} = \frac{1}{2} \times \text{diagonal}^2 \]
Given the diagonal of the square is 32 cm, we substitute this value into the formula:
\[ \text{Area of square} = \frac{1}{2} \times (32)^2 = \frac{1}{2} \times 1024 = 512 \text{ cm}^2 \]

### Step 2: Calculate the area of the equilateral triangle
The area of an equilateral triangle can be calculated using the formula:
\[ \text{Area of triangle} = \frac{\sqrt{3}}{4} \times a^2 \]
where \( a \) is the length of the side of the triangle. Given that the base of the triangle is 8 cm, we can use this as the side length:
\[ \text{Area of triangle} = \frac{\sqrt{3}}{4} \times (8)^2 = \frac{\sqrt{3}}{4} \times 64 = 16\sqrt{3} \text{ cm}^2 \]
Now, substituting \(\sqrt{3} = 1.732\):
\[ \text{Area of triangle} = 16 \times 1.732 = 27.712 \text{ cm}^2 \]

### Step 3: Calculate the total area
Now, we can find the total area of the kite by adding the area of the square and the area of the triangle:
\[ \text{Total area} = \text{Area of square} + \text{Area of triangle} = 512 + 27.712 = 539.712 \text{ cm}^2 \]

### Final Answer
Thus, the total area of paper used to make the kite is approximately:
\[ \boxed{539.712 \text{ cm}^2} \]

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