Question

The area of circle inscribed in an equilateral triangle is 48 square units Then the perimeter of the triangle is ( in units)

• A. $72\sqrt{3}$
• B. $48\sqrt{3}$
• C. 72
• D. 36

Answer 1

Answer: (C)

72

The area of circle 48

Then $\pi r^2=48\Rightarrow r^2=48\Rightarrow r=4\sqrt{\frac{3}{\pi }}$

The center of the circle is also the in center of the triangle. Draw the three altitudes of the triangle. They are concurrent at the in center (which is also the or tho center), and they dissect the triangle into six congruent right triangles. They are ${30}^0-{60}^0-{90}^0$ triangles, each having short leg r. Let x be the length of the long leg.

$x=rtan{60}^0=4\sqrt{\frac{3}{\pi }}\times \sqrt{3}=\frac{12}{\sqrt{\pi }}$

The measure x is also half the length of a side of the given equilateral triangle, which makes it one-sixth the perimeter.

Perimeter of triangle $6x=6\times \frac{12}{\sqrt{\pi }}=\frac{72}{\pi }$