If the perimeter of an equilateral triangle is 36 cm, calculate its area and height.

We know that,

Perimeter of an equilateral triangle = 3 × side.

Substituting the values,

⇒ 36 = 3 × side

⇒ side = $\dfrac{36}{3}$ = 12 cm.

Area of equilateral triangle = $\dfrac{\sqrt{3}}{4}(side)^2$

Substituting the values we get,

$A = \dfrac{\sqrt{3}}{4} \times (12)^2 \\[1em] = \dfrac{\sqrt{3}}{4} \times 144 \\[1em] = 36\sqrt{3} \\[1em] = 36 \times 1.732 \\[1em] = 62.4 \text{ cm}^2$

From figure,

In triangle ABD,

Using Pythagoras Theorem,

AB² = AD² + BD² …….(1)

The perpendicular from a vertex of an equilateral triangle to the opposite side, bisects it.

So, BD = $\dfrac{12}{2}$ = 6 cm.

Substituting the values in (1) we get,

⇒ 12² = AD² + 6²

⇒ 144 = AD² + 36

⇒ AD² = 144 – 36 = 108

⇒ AD = $\sqrt{108}$ = 10.4 cm.

Hence, area of triangle = 62.4 cm² and height = 10.4 cm.