To find the area of a regular hexagon, we can use the following formula:

$\text{Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem}$

Where:

• The Perimeter is the total length of all sides of the hexagon.
• The Apothem is the distance from the center of the hexagon to the midpoint of a side.

Given:

• Apothem = 5.2 cm
• Side length = 6 cm

Step 1: Find the perimeter of the hexagon.

Since a regular hexagon has 6 equal sides, the perimeter $P$ is:

$P = 6 \times 6 = 36 \, \text{cm}$

Step 2: Use the area formula.

Now, substitute the values of the perimeter and the apothem into the area formula:

$\text{Area} = \frac{1}{2} \times 36 \, \text{cm} \times 5.2 \, \text{cm}$

$\text{Area} = \frac{1}{2} \times 187.2 \, \text{cm}^2$

$\text{Area} = 93.6 \, \text{cm}^2$

Final Answer:

The area of the regular hexagon is 93.6 cm², rounded to the nearest tenth.

Would you like more details on any part of this solution? Here are 5 related questions to further explore:

1. How does the apothem relate to the area of a regular polygon?
2. What other methods can be used to find the area of a regular hexagon?
3. How would the area change if the side length was doubled but the apothem remained the same?
4. Can you find the area of the hexagon using trigonometry instead of the apothem formula?
5. How do you calculate the perimeter of a regular polygon with a different number of sides?

Tip: The apothem is especially useful for finding the area of regular polygons because it simplifies the formula to a more practical calculation.