To find the area of a regular hexagon given its apothem and side length, we can follow these steps:
1. Understand the components:
- The apothem (denoted as [tex]\(a\)[/tex]) is the perpendicular distance from the center of the hexagon to the midpoint of one of its sides. In this case, the apothem is 5.2 cm.
- The side length (denoted as [tex]\(s\)[/tex]) is the length of one side of the hexagon. In this case, the side length is 6 cm.
2. Calculate the Perimeter:
- A regular hexagon has six equal sides. Therefore, to find the perimeter ([tex]\(P\)[/tex]) of the hexagon, we multiply the side length by 6.
[tex]\[
P = 6 \times s = 6 \times 6 = 36 \text{ cm}
\][/tex]
3. Use the area formula:
- The area ([tex]\(A\)[/tex]) of a regular hexagon can be calculated using the formula involving the perimeter and the apothem:
[tex]\[
A = \frac{1}{2} \times P \times a
\][/tex]
4. Plug in the values:
- Substitute the calculated perimeter and the given apothem into the formula.
[tex]\[
A = \frac{1}{2} \times 36 \times 5.2
\][/tex]
5. Calculate the area:
- Perform the multiplication in steps:
[tex]\[
\frac{1}{2} \times 36 = 18
\][/tex]
[tex]\[
18 \times 5.2 = 93.6
\][/tex]
Hence, the area of the regular hexagon is [tex]\(93.6 \text{ cm}^2\)[/tex].
