+4609 Triangle ABC is an equilateral triangle and O is the center of its inscribed circle. If the area of the circle is \(4\pi\) sq cm, what is the area, in square centimeters, of triangle ABC? Express your answer in simplest radical form
+128474 The radius of the circle can be found as
4pi = pi*r^2 ⇒ r = 2
(1/2) of the side, m, of the triangle can be found as
2 / sin(30) = m / sin( 60)
2sin(60) / sin( 30) = m
4 * sin 60 =
4 *√3/2 = m = 2√3
So.....the side of the triangle is twice this = 4√3 = √48
And the area is
(1/2)( √48)^2 sin 60 =
24 * √3 / 2
12√3 cm^2
