Find the smallest possible area of an isosceles triangle that is circumscribed about a circle of radius $r = 4$.
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 with an incircle of radius $ r $, the side length $ s $ is $ 2\sqrt{3}r $. The area $ A $ is given by $ \frac{\sqrt{3}}{4} s^2 $, which simplifies to $ 48\sqrt{3} $ when $ r = 4 $.

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

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