What is the area of one of the sectors formed by the radii to the vertices of the triangle?

Question

An equilateral triangle with side 2root3 is inscribed in a circle. What is the area of one of the sectors formed by the radii to the vertices of the triangle? explain how you got answer.

Answer 1

Envision bisecting all 3 angles of the equilateral triangle.  The angle bisectors also bisect the opposite sides.  You end up with (6) equal 30-60-90 triangles having the side proportions 1, 2, √3.

The 2-unit side is also the radius of the circle in which the triangle is inscribed.  The three radii divide the circle into 3 equal sectors.  We know the area of the complete circle is πr^2, or in this case π(2^2) = 4π. 1/3 of this area is (4/3)π, which is about 1.04719755 square units.