Question

A circle with area 36\(\pi\) sq. cm is inscribed in an equilateral triangle. What is the area (in sq.cm) of the triangle?

• A. \(27\sqrt3\)
• B. \(54\sqrt3\)
• C. \(108\sqrt3\)
• D. \(144\sqrt3\)

Answer 1

The Correct Option is C

A circle is inscribed in an equilateral triangle, and the area of the circle is:

\[ \pi r^2 = 36\pi \]

Step 1: Find the Radius of the Circle 

Dividing both sides by \( \pi \):

\[ r^2 = 36 \]

\[ r = 6 \]

Step 2: Use the Formula for the Inradius of an Equilateral Triangle

For an equilateral triangle with side length \( s \), the inradius \( r \) is given by:

\[ r = \frac{s \sqrt{3}}{6} \]

Substituting \( r = 6 \):

\[ 6 = \frac{s \sqrt{3}}{6} \]

Multiplying both sides by 6:

\[ s \sqrt{3} = 36 \]

\[ s = \frac{36}{\sqrt{3}} \]

Rationalizing the denominator:

\[ s = \frac{36 \times \sqrt{3}}{3} = 12\sqrt{3} \]

Step 3: Compute the Area of the Equilateral Triangle

The formula for the area of an equilateral triangle is:

\[ A = \frac{\sqrt{3}}{4} s^2 \]

Substituting \( s = 12\sqrt{3} \):

\[ A = \frac{\sqrt{3}}{4} (12\sqrt{3})^2 \]

\[ A = \frac{\sqrt{3}}{4} \times (144 \times 3) \]

\[ A = \frac{\sqrt{3} \times 432}{4} \]

\[ A = \frac{432\sqrt{3}}{4} \]

\[ A = 108 \sqrt{3} \]

Final Answer:

\( 108 \sqrt{3} \) sq. cm