A circle is inscribed in an equilateral triangle with side length 12 cm. What is the radius of the circle?
A circle is inscribed in an equilateral triangle with side length 12 cm. What is the radius of the circle?
Show Hint
- $2\sqrt{3}$ cm
- $3\sqrt{3}$ cm
- $4\sqrt{3}$ cm
$6\sqrt{3}$ cm
The Correct Option is A
Solution and Explanation
- 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 $r = \frac{\text{Area}}{s} = \frac{36\sqrt{3}}{18} = 2\sqrt{3}$.
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