Question

An equilateral triangle and a regular hexagon are inscribed in the same circle such that the three vertices of the triangle coincide with three of the vertices of the hexagon. If the side length of the equilateral triangle is 10 radical 3 cm, find the area of the part of the interior of the regular hexagon that is not in the interior of the triangle.

Answer 1

Draw a sketch, and you know the internal angles of the hexagon are 120 degrees each (720/6) so you get three isosceles triangles with top angle 120 degrees and long side 10 radical 3 cm. Split one of the isosceles triangles into right-angled triangles and use pythagoras to find the height, then work out the area.

Answer 2

Sorry not Pythagoras, use trigonometry to find the height (I used tan, that way you don't need the hypotenuse).