To find the area of a regular hexagon with an apothem of [tex]\( 3.5 \, \text{cm} \)[/tex] and a side length of [tex]\( 4 \, \text{cm} \)[/tex], follow these steps:
1. Determine the perimeter of the hexagon:
- A regular hexagon has 6 sides.
- Each side length is [tex]\( 4 \, \text{cm} \)[/tex].
- The perimeter [tex]\( P \)[/tex] of the hexagon is calculated by multiplying the number of sides by the length of each side:
[tex]\[
P = 6 \times 4 \, \text{cm} = 24 \, \text{cm}
\][/tex]
2. Use the formula for the area of a regular hexagon:
- The area [tex]\( A \)[/tex] of a regular hexagon can be found using the formula:
[tex]\[
A = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem}
\][/tex]
- Substitute the known values into the formula:
[tex]\[
A = \frac{1}{2} \times 24 \, \text{cm} \times 3.5 \, \text{cm}
\][/tex]
3. Perform the multiplication:
- First, multiply the perimeter by the apothem:
[tex]\[
24 \, \text{cm} \times 3.5 \, \text{cm} = 84 \, \text{cm}^2
\][/tex]
- Then, multiply the result by [tex]\( \frac{1}{2} \)[/tex]:
[tex]\[
\frac{1}{2} \times 84 \, \text{cm}^2 = 42 \, \text{cm}^2
\][/tex]
Thus, the area of the regular hexagon is [tex]\( 42 \, \text{cm}^2 \)[/tex].
