A triangle has sides of length 5, 6, and 7. Use Heron's formula to find the area.

Heron's formula for the area of a triangle with sides $$a, b, c$$ is: Area $$= \sqrt{s(s-a)(s-b)(s-c)}$$, where $$s$$ is the semi-perimeter, $$s = \frac{a+b+c}{2}$$.

Given:

• $$a = 5$$
• $$b = 6$$
• $$c = 7$$

First, calculate the semi-perimeter:

\[ s = \frac{5+6+7}{2} = \frac{18}{2} = 9 \]

Now, apply Heron's formula:

\[ \text{Area} = \sqrt{9(9-5)(9-6)(9-7)} \]

\[ \text{Area} = \sqrt{9(4)(3)(2)} \]

\[ \text{Area} = \sqrt{9 \times 24} \]

\[ \text{Area} = \sqrt{216} \]

\[ \text{Area} = \sqrt{36 \times 6} = 6\sqrt{6} \]

Ответ: $$6\sqrt{6}$$