Questions: Find the area of this triangular field to the nearest hectare (ha).

Solution Steps

The sides of the triangle are given as:

• AB = 314 m
• BC = 238 m
• AC = 407 m

The semi-perimeter \( s \) of the triangle is calculated using the formula: \[ s = \frac{a + b + c}{2} \] where \( a = 314 \) m, \( b = 238 \) m, and \( c = 407 \) m. \[ s = \frac{314 + 238 + 407}{2} = 479.5 \text{ m} \]

Heron's formula for the area \( A \) of a triangle is: \[ A = \sqrt{s(s-a)(s-b)(s-c)} \] Substitute the values: \[ A = \sqrt{479.5 \times (479.5 - 314) \times (479.5 - 238) \times (479.5 - 407)} \] \[ A = \sqrt{479.5 \times 165.5 \times 241.5 \times 72.5} \] \[ A = \sqrt{479.5 \times 165.5 \times 241.5 \times 72.5} \approx \sqrt{137,019,000} \approx 11,704.6 \text{ m}^2 \]

1 hectare (ha) = 10,000 square meters (m²). \[ \text{Area in hectares} = \frac{11,704.6 \text{ m}^2}{10,000} \approx 1.17046 \text{ ha} \]

\[ \text{Area} \approx 1 \text{ ha} \]

Final Answer