12. Find the side length and area of an equilateral triangle with apothem 3 cm .

To find the side length of an equilateral triangle with an apothem of 3 cm, you can use the relationship between the apothem (a), the side length (s), and the formula for the apothem of an equilateral triangle: \( a = \frac{s \sqrt{3}}{6} \). Solving for s gives us \( s = \frac{6a}{\sqrt{3}} = \frac{6 \times 3}{\sqrt{3}} = 6\sqrt{3} \approx 10.39 \) cm. Next, the area (A) of the triangle can be found using the formula \( A = \frac{1}{2} \times \text{perimeter} \times a \). The perimeter \( P \) of the triangle is \( 3s = 3(6\sqrt{3}) = 18\sqrt{3} \). Thus, \( A = \frac{1}{2} \times 18\sqrt{3} \times 3 = 27\sqrt{3} \approx 46.76 \text{ cm}^2 \).