Question

The ratio of the areas of the incircle and the circumcircle of an equilateral triangle is: 

1. 1 : 2
2. 1 : 4
3. 1 : 8
4. 3 : 2

Answer 1

Correct Answer - Option 2 : 1 : 4

Given:

We have to find the ratio of the areas of the incircle and the circumcircle of an equilateral triangle.

Formula used:

The radius of incentre = a/(2√3)

The radius of circumcentre = a/√3

Area of circle = πr²

Where,

a = side of an equilateral triangle

r = radius of the circle

Calculation:

Let the side of the triangle be ‘a’.

The radius of incentre = a/(2√3)

The radius of circumcentre = a/√3

Area of incircle = π[a/(2√3)]²

Area of circumcircle = π[a/(√3)]²

\( \Rightarrow \;\frac{{{\rm{Area\;of\;incircle}}}}{{{\rm{Area\;of\;circumcircle}}}}\; = \;\frac{{{\rm{\pi }}{{\left[ {{\rm{a}}/\left( {2\surd 3} \right)} \right]}^2}}}{{{\rm{\pi }}{{\left[ {{\rm{a}}/\left( {\surd 3} \right)} \right]}^2}}}\)

\( \Rightarrow \frac{{{\rm{\pi }}{{\left[ {{\rm{a}}/\left( {2\surd 3} \right)} \right]}^2}}}{{{\rm{\pi }}{{\left[ {{\rm{a}}/\left( {\surd 3} \right)} \right]}^2}}}\)

⇒ 1 : 4

∴ The ratio is 1 : 4.