Find the perimeter and area of an equilateral triangle whose height is 12 cm. Write your answers, correct to two decimal places.

Given,

Height (h) = 12 cm

Let the length of the side of an equilateral triangle be a cm.

We know that for an equilateral triangle,

$\Rightarrow \text{Height} = \dfrac{\sqrt{3}}{2}a \\[1em] \Rightarrow 12 = \dfrac{\sqrt{3}}{2}a \\[1em] \Rightarrow a = \dfrac{12 \times 2}{\sqrt{3}} = \dfrac{24}{\sqrt{3}} \\[1em] \Rightarrow a = \dfrac{24\sqrt{3}}{3} = 8\sqrt{3} \text{ cm}$

Perimeter of an equilateral triangle = 3 × side

= 3 × 8 $\sqrt{3}$

= 3 × 8 × 1.732

= 41.568 ≈ 41.57 cm.

$\Rightarrow \text{Area of an equilateral triangle} = \dfrac{\sqrt{3}}{4} \times (side)^2 \\[1em] \Rightarrow \dfrac{\sqrt{3}}{4} \times (8 \sqrt{3})^2 \\[1em] \Rightarrow \dfrac{\sqrt{3}}{4} \times 64 \times 3 \\[1em] \Rightarrow \sqrt{3} \times 48 \\[1em] \Rightarrow 1.732 \times 48 \\[1em] \Rightarrow 83.136 \approx 83.14 \text{ cm}^2. \\[1em]$

Hence, the perimeter = 41.57 cm and area = 83.14 cm².