Regular Hexagons: Applying the formula

The formula to find the area of a regular hexagon with side length $s$ is given by:

$\frac{3 \sqrt{3}}{2} s^2$

For a hexagon with side length $10 \text{ cm}$, substitute $s = 10$ into the formula:

$\text{Area} = \frac{3 \sqrt{3}}{2} \times 10^2$

Calculate $10^2$:

$10^2 = 100$

Substitute back:

$\text{Area} = \frac{3 \sqrt{3}}{2} \times 100$

This simplifies to:

$259.81 \text{ cm}^2$

The formula to find the area of a regular hexagon with side length $s$ is given by:

$\frac{3 \sqrt{3}}{2} s^2$

For a hexagon with side length $8 \text{ cm}$, substitute $s = 8$ into the formula:

$\text{Area} = \frac{3 \sqrt{3}}{2} \times 8^2$

Calculate $8^2$:

$8^2 = 64$

Substitute back:

$\text{Area} = \frac{3 \sqrt{3}}{2} \times 64$

This simplifies to:

$166.28 \text{ cm}^2$

To solve this problem, we'll follow these steps:

• Step 1: Identify the given information in the drawing.
• Step 2: Apply the formula for the area of a regular hexagon.
• Step 3: Perform the necessary calculations using available data.

Now, let's work through each step:

Step 1: From the drawing, a side length of "4" is provided, indicated next to a blue segment. Assuming this corresponds to the side length of the regular hexagon.

Step 2: We'll use the formula for the area of a regular hexagon, which is $\frac{3\sqrt{3}}{2} \times s^2$.

Step 3: Plugging in the side length $s = 4$, our calculation is:

Approximating $\sqrt{3} \approx 1.732$, we have:

Rounding this value gives approximately 41.56, which matches the given correct answer choice $41.56$.

To determine the area of the hexagon, follow these steps:

• Step 1: Recognize the provided numeral "9" as representing the diameter of the hexagon.
• Step 2: Convert the diameter to a radius by dividing by 2. Thus, the radius $r = \frac{9}{2} = 4.5$.
• Step 3: Use the regular hexagon area formula with the radius: $A = \frac{3\sqrt{3}}{2} r^2$.

Now, we compute the area using the formula:
Step 3: Plugging in the radius, we have:
$A = \frac{3\sqrt{3}}{2} (4.5)^2$.

First, calculate $(4.5)^2 = 20.25$.
Now calculate $A = \frac{3\sqrt{3}}{2} \times 20.25$.
The approximate value of $\sqrt{3}$ is 1.732. Continue the calculation:
$A = \frac{3 \times 1.732}{2} \times 20.25 \approx \frac{5.196}{2} \times 20.25 \approx 2.598 \times 20.25 \approx 52.61$.

This calculation contrives that the area calculation changed after correction:
The area of the hexagon is $\approx 52.61$.

Therefore, the correct choice is the area: $52.61$.

To solve this problem, we'll proceed with the following steps:

• Step 1: Interpreting the given measure as the diameter of the hexagon and finding the radius.
• Step 2: Calculate the side length using the radius.
• Step 3: Determine the area using the formula for a regular hexagon.

Here's the detailed process:

Step 1: The given length of 14 is the diameter of the hexagon, which means the diagonal from one vertex, passing through the center, to the opposite vertex. The radius ($r$) from the center to a vertex would be half of this diameter:

$r = \frac{14}{2} = 7$

Step 2: The side length $s$ for a regular hexagon is related to the radius by the central triangle consistency (equilateral triangles formed by connecting the center). The side length is equal to the radius:

$s = r = 7$

Step 3: Now, use the formula for the area of a regular hexagon:

$\text{Area} = \frac{3\sqrt{3}}{2} s^2$

Substitute the side length calculated:

$\text{Area} = \frac{3\sqrt{3}}{2} \cdot 7^2$

$\text{Area} = \frac{3\sqrt{3}}{2} \cdot 49$

$\text{Area} = \frac{3\sqrt{3}}{2} \cdot 49 \approx 127.3$

Therefore, the area of the hexagon is approximately $\mathbf{127.3}$. This matches choice 2 in the list of possible answers.