Question

A circle is inscribed in an equilateral triangle with side length 12 cm. What is the radius of the circle? 
 

• A. $2\sqrt{3}$ cm
• B. $3\sqrt{3}$ cm
• C. $4\sqrt{3}$ cm
• D. $6\sqrt{3}$ cm

Hint:

For an equilateral triangle, memorize $r = \frac{a\sqrt{3}}{6}$ for quick inradius calculation.

Answer 1

The Correct Option is A

- Step 1: Inradius formula for equilateral triangle - $r = \frac{a\sqrt{3}}{6}$. 

- Step 2: Substitute - $a=12$: \[ r = \frac{12\sqrt{3}}{6} = 2\sqrt{3} \ \text{cm} \] 

- Step 3: Alternate check using area and semiperimeter - 

Area = $\frac{\sqrt{3}}{4} \times 12^2 = 36\sqrt{3}$ cm². 

Semiperimeter $s = \frac{3\times 12}{2} = 18$ cm. 

Then $r = \frac{\text{Area}}{s} = \frac{36\sqrt{3}}{18} = 2\sqrt{3}$.