To find the area of a triangle when given the lengths of all three sides, you can use Heron's formula. The sides of the triangle are \(a = 16 , \text{ft}\), \(b = 12 , \text{ft}\), and \(c = 14 , \text{ft}\).
-
Calculate the semi-perimeter (s): \[ s = \frac{a + b + c}{2} = \frac{16 + 12 + 14}{2} = \frac{42}{2} = 21 , \text{ft} \]
-
Now apply Heron's formula to find the area (A): \[ A = \sqrt{s(s - a)(s - b)(s - c)} \]
Plug in the values: \[ A = \sqrt{21(21 - 16)(21 - 12)(21 - 14)} \] \[ A = \sqrt{21 \times 5 \times 9 \times 7} \]
-
Calculate each part: \[ A = \sqrt{21 \times 5 \times 9 \times 7} \] First, calculate the products:
- \(21 \times 5 = 105\)
- \(9 \times 7 = 63\)
Now multiply these results: \[ 105 \times 63 = 6615 \]
-
Take the square root: \[ A = \sqrt{6615} \approx 81.3 , \text{ft²} \]
Thus, the area of the triangle is approximately \( \boxed{81.3} , \text{ft}^2 \).