To find the area of an equilateral triangle using its apothem and perimeter, we can use the formula for the area of a regular polygon:
[tex]\[ \text{Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem} \][/tex]
Given:
- Apothem ([tex]\(a\)[/tex]) = 2.16 cm
- Perimeter ([tex]\(P\)[/tex]) = 22.45 cm
First, we plug these values into the formula:
[tex]\[ \text{Area} = \frac{1}{2} \times 22.45 \, \text{cm} \times 2.16 \, \text{cm} \][/tex]
Next, we calculate the product of the perimeter and the apothem:
[tex]\[ 22.45 \times 2.16 = 48.092 \][/tex]
Now, we divide this product by 2 to find the area of the triangle:
[tex]\[ \text{Area} = \frac{48.092}{2} = 24.246 \, \text{cm}^2 \][/tex]
Finally, we round this result to the nearest tenth:
[tex]\[ 24.246 \, \text{cm}^2 \approx 24.2 \, \text{cm}^2 \][/tex]
Thus, the area of the equilateral triangle, rounded to the nearest tenth, is:
[tex]\[ 24.2 \, \text{cm}^2 \][/tex]
Therefore, the correct answer is [tex]\(24.2 \, \text{cm}^2\)[/tex].
