An isosceles triangle has perimeter 30 cm and each of the equal sides is 12 cm. Find the area of the triangle.

Length of equal sides (a and b) = 12 cm

Let third side of triangle (c) = x cm

Given,

Perimeter of triangle = 30

∴ 12 + 12 + x = 30

⇒ 24 + x = 30

⇒ x = 30 - 24

⇒ x = 6 cm.

∴ c = 6 cm.

By formula,

Semi perimeter (s) = $\dfrac{\text{Perimeter of triangle}}{2} = \dfrac{30}{2}$ = 15 cm.

By Heron's formula,

Area of triangle (A) = $\sqrt{s(s - a)(s - b)(s - c)}$ sq.units

Substituting values we get :

$A = \sqrt{15(15 - 12)(15 - 12)(15 - 6)} \\[1em] = \sqrt{15 \times 3 \times 3 \times 9} \\[1em] = \sqrt{1215} \\[1em] = 9\sqrt{15} \text{ cm}^2.$

Hence, area of triangle = $9\sqrt{15}$ cm².