Question

In $\Delta$ $ABC$ the sides opposite to angles $A,B,C$ are denoted by $a,b,c$ respectively.

In an equilateral triangle, the inradius$\left(r\right)$ and the circumradius$\left(R\right)$ are connected by which of the following?

• A. $r=4R$
• B. $r=\frac{R}{2}$
• C. $r=\frac{R}{3}$
• D. None of these

Answer 1

The correct option is B $r=\frac{R}{2}$

Let side of equilateral triangle is $m$

We know $R=\frac{abc}{4\Delta }$

$a=b=c=m$ and $\Delta =\frac{\sqrt{3}{a}^2}{4}$

Hence $R=\frac{m^3}{\frac{4\sqrt{3}{m}^2}{4}}$

$R=\frac{m}{\sqrt{3}}$ - - - - - - - - - $1$

We know $inradius\left(r\right)=\frac{\Delta }{s}$

$s=\frac{a+b+c}{2}$

$r=\frac{\sqrt{3}{m}^2}{4s}$ - - - - - - - - $2$

Put the value of s in equ($2$)

We get $r=\frac{m}{2\sqrt{3}}$ - - - - - - $3$

Now compare equ($1$) and equ($3$)

We will get $R=2r$