Question

In an equilateral triangle the inradius and the circumradius are connected by

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

Answer 1

Answer: (B)

$r = \frac { R } { 2 }$

Answer 2

$r = 4 R \sin \frac { A } { 2 } \cdot \sin \frac { B } { 2 } \cdot \sin \frac { C } { 2 }$

For an equilateral triangle, $A = B = C = 60 ^ { \circ }$

$\therefore$ $r = 4 R \sin 30 ^ { \circ } \cdot \sin 30 ^ { \circ } \cdot \sin 30 ^ { \circ }$ $= 4 R \cdot \frac { 1 } { 2 } \cdot \frac { 1 } { 2 } \cdot \frac { 1 } { 2 }$ $= \frac { R } { 2 }$.