Find the smallest possible area of an isosceles triangle that is circumscribed about a circle of radius $r = 4$. Select the correct answer.
A) $4 \pi \sqrt { 3 }$
B) $4 \sqrt { 3 }$
C) $48 \sqrt { 3 }$
D) $\sqrt { 3 }$
E) $\sqrt { 48 } r ^ { 2 }$

Question

Quizplus · Accepted Answer

The smallest area of an isosceles triangle circumscribed about a circle is achieved when the triangle is equilateral. For an equilateral triangle, the area $A$ is given by $A = \frac{\sqrt{3}}{4} s^2$, where $s$ is the side length. The radius $r$ of the inscribed circle is $\frac{s \sqrt{3}}{6}$. Solving for $s$ when $r = 4$, we get $s = 8\sqrt{3}$. Substituting $s$ into the area formula gives $A = \frac{\sqrt{3}}{4} (8\sqrt{3})^2 = 48\sqrt{3}$.

Find the Smallest Possible Area of an Isosceles Triangle That r=4r = 4r=4

Find the Smallest Possible Area of an Isosceles Triangle That $r = 4$