Triangle ABC is an equilateral triangle and O is the center of its inscribed circle. If the area of the circle is 16*pi sq cm, what is the area, in square centimeters, of triangle ABC? Express your answer in simplest radical form.
1) Find the radius of the inscribed circle by using the formula: Area = pi·r2.
2) Find the length of each side of the equilateral triangle by using the formula:
radius of the incircle = side of the equilateral triangle / 3
3) Find the area of the equilateral triangle by using the formula: Area = ( sqrt(3) / 4 ) · side of equilateral triangle
Area of circle = pi(r^2) = 16pi => r = 4 cm.
ABC is equilateral, A = B = C = 600.
Consider circle center O inscribed in ABC: from center O drop a perpendicular, which is the radius r to base of triangle BC at M
which is the midpoint of BC and from O joined to B. This forms another right triangle OBM where B = 300, O = 600 and M = 900.
OM = r = 4 cm, BM = 4/(tan 300) = 4/(1/√3) = 4√3.
Length of BC = 2(BM) = 8√3.
Area of ABC = (1/2)(8√3)(8√3)(sin 600) = (1/2)(8√3)(8√3)(√3/2) = (192√3)/4 = 48√3 sq cm.