Question

The length of side of an equilateral triangle is $12$ cm. Find the ratio of the areas of the circle circumscribing the triangle to the circle inscribing the triangle

• A. $2:1$
• B. $5:4$
• C. $4:1$
• D. $3:2$

Answer 1

Answer: (C)

$4:1$

We have;

Length of side of an equilateral triangle $a=b=c=12cm$

Circumradius $R=\frac{abc}{4\Delta }$

$\Delta =\frac{\sqrt{3}}{4}{a}^2=\frac{\sqrt{3}}{4}\times 144=36\sqrt{3}c{m}^2$

$R=\frac{12\times 12\times 12}{4\times \sqrt{3}\times 36}=4\sqrt{3}cm$

$S=\frac{a+b+c}{2}=\frac{36}{2}=18cm$

In radius $r=\frac{\Delta }{S}=\frac{36\sqrt{3}}{18}=2\sqrt{3}cm$

Now;

$\frac{\text{area of circle circumscribing the triangle}}{\text{Area of circle inscribing the triangle}}$

$\frac{{\pi R}^2}{{\pi r}^2}=\frac{R^2}{r^2}=\frac{16\times 3}{4\times 3}=4:1$

Hence, the ratio is $4:1$