what is the area of a hexagon if: one side = 12, apothem =10.4

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

\[
A = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem}
\]

### Steps:

1. **Calculate the Perimeter**:
   Since a regular hexagon has 6 equal sides, the perimeter \(P\) is:
   \[
   P = 6 \times \text{side length} = 6 \times 12 = 72 \text{ units}
   \]

2. **Use the Area Formula**:
   Now, plug the values into the area formula:
   \[
   A = \frac{1}{2} \times 72 \times 10.4
   \]
   \[
   A = 36 \times 10.4 = 374.4 \text{ square units}
   \]

### Final Answer:
- **Area:** \( 374.4 \) square units

Would you like more details or have any questions?

Here are 5 related questions you might find interesting:
1. How would the area change if the side length of the hexagon were doubled?
2. Can you find the area of a regular hexagon using only the side length?
3. How does the apothem relate to the radius of the hexagon's circumcircle?
4. What is the perimeter of the hexagon if the side length is increased to 15 units?
5. How do you find the area of an irregular hexagon?

**Tip:** The apothem is always perpendicular to a side of the hexagon and is useful for calculating the area, especially when only the side length and apothem are known.

Learn how to calculate the area of a regular hexagon with a side length of 12 units and an apothem of 10.4 units. Follow a detailed step-by-step solution using the area formula for regular hexagons. Suitable for high school students.

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