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 possible area of an isosceles triangle circumscribed about a circle (inscribed in the triangle) occurs when the triangle is equilateral. The side length of such an equilateral triangle can be found using the formula for the radius (r) of the circle inscribed in an equilateral triangle, which is $r = \frac{s\sqrt{3}}{6}$, where $s$ is the side length of the triangle. Rearranging this formula to solve for $s$ gives $s = \frac{6r}{\sqrt{3}} = 2\sqrt{3}r$. Substituting $r = 4$ gives $s = 8\sqrt{3}$. The area of an equilateral triangle is given by the formula $\frac{\sqrt{3}}{4}s^2$, substituting $s = 8\sqrt{3}$ into this formula gives an area of $48\sqrt{3}$.

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

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