Calculating Hexagon Area Inscribed in a Circle
The problem asks for the area of a regular hexagon inscribed inside a circle with a given radius.
Key Information
- Shape: Regular Hexagon
- Inscribed in: Circle
- Circle Radius ($r$): 14 cm
Geometric Properties Used
A regular hexagon can be divided into 6 identical equilateral triangles. The side length ($s$) of each equilateral triangle is equal to the radius ($r$) of the circle in which the hexagon is inscribed.
Step-by-Step Calculation
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Determine Side Length: Since the hexagon is inscribed in the circle, its side length ($s$) is equal to the circle's radius.
$s = r = 14 \text{ cm}$
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Area of Equilateral Triangle: The formula for the area of an equilateral triangle with side length $s$ is:
$\text{Area}_{\triangle} = \frac{\sqrt{3}}{4} s^2$
Substitute $s = 14$ cm:$\text{Area}_{\triangle} = \frac{\sqrt{3}}{4} (14 \text{ cm})^2 = \frac{\sqrt{3}}{4} \times 196 \text{ cm}^2 = 49\sqrt{3} \text{ cm}^2$
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Area of Hexagon: The area of the hexagon is 6 times the area of one equilateral triangle.
$\text{Area}_{\text{Hexagon}} = 6 \times \text{Area}_{\triangle} = 6 \times 49\sqrt{3} \text{ cm}^2 = 294\sqrt{3} \text{ cm}^2$
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Approximate Value: Use the approximate value of $\sqrt{3} \approx 1.73205$.
$\text{Area}_{\text{Hexagon}} \approx 294 \times 1.73205 \text{ cm}^2 \approx 509.2237 \text{ cm}^2$
Conclusion
The calculated area is approximately $509.22 \text{ cm}^2$. Comparing this with the given options, Option B ($509.21 \text{ cm}^2$) is the closest value.
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