Find the area of the triangle whose sides are 30 cm, 24 cm and 18 cm. Also, find the length of the altitude corresponding to the smallest side of the triangle.

Let a = 30 cm, b = 24 cm, c = 18 cm.

Then,

s = $\dfrac{1}{2}(a + b + c) = \dfrac{1}{2}(30 + 24 + 18) = \dfrac{72}{2}$ = 36 cm.

⇒ (s - a) = (36 - 30) cm = 6 cm.

⇒ (s - b) = (36 - 24) cm = 12 cm.

⇒ (s - c) = (36 - 18) cm = 18 cm.

We know that,

$\Rightarrow \text{Area of triangle} = \sqrt{s(s - a)(s - b)(s - c)} \\[1em] \Rightarrow \sqrt{36 × 6 × 12 × 18} \\[1em] \Rightarrow \sqrt{46656} \\[1em] \Rightarrow 216 \text{ cm}^2. \\[1em]$

Length of the smallest side = 18 cm.

Let the length of altitude be x cm. Then,

$\Rightarrow Area = \dfrac{1}{2} \times \text{ Base } \times \text{ Altitude } \\[1em] \Rightarrow 216 = \dfrac{1}{2} \times 18 \times x \\[1em] \Rightarrow 216 = 9x \\[1em] \Rightarrow x = \dfrac{216}{9} \\[1em] \Rightarrow 24 \text{ cm}. \\[1em]$

Hence, area of triangle = 216 cm² and altitude = 24 cm.