Triangle Area Calculation Using Medians

The area of a triangle can be determined if the lengths of its three medians are known. A specific formula relates the area to the median lengths.

Median Area Formula

Let the lengths of the medians be $m_a, m_b, m_c$. The area ($A$) of the triangle is given by:

$ A = \frac{4}{3} \sqrt{s_m (s_m - m_a) (s_m - m_b) (s_m - m_c)} $

where $s_m$ is the semi-perimeter of a triangle whose sides are the lengths of the medians:

$ s_m = \frac{m_a + m_b + m_c}{2} $

Problem Calculation

Given median lengths: $m_a = 3$ cm, $m_b = 6$ cm, $m_c = 5$ cm.

Step 1: Calculate Semi-Perimeter ($s_m$)

Calculate $s_m$ using the given median lengths:

$ s_m = \frac{3 + 6 + 5}{2} = \frac{14}{2} = 7 \text{ cm} $

Step 2: Calculate Triangle Area ($A$)

Substitute the values into the area formula:

$ A = \frac{4}{3} \sqrt{7 (7 - 3) (7 - 6) (7 - 5)} $

$ A = \frac{4}{3} \sqrt{7 \times 4 \times 1 \times 2} $

$ A = \frac{4}{3} \sqrt{56} $

The area of the triangle is $\frac{4}{3} \sqrt{56} \text{ cm}^2$.