UNIT circle question

Question

An equilateral triangle is inscribed in the unit circle. find the exact perimeter of the equilateral triangle.

Answer 1

Consider the angular bisectors of the inscribed triangle.  Those bisectors meet at the center of the triangle.   Any two of them along with a side of the triangle form a smaller isosceles triangle with base angles = 30 degrees and sides equal to the radius of the unit circle, which is 1 of course.

 

The remaining angle of that isosceles triangle is 180 - 2*30 = 120.   We can then calculate the base of the isosceles triangle using the law of sines.

 

sin 30/1  = sin 120 /x    => x = sin 120/sin 30 = sqrt 3;

 

Since this base is one side of an equilateral triangle, the perimeter of the triangle is then 3 * sqrt 3.

Answer 2

See

https://www.wyzant.com/resources/answers/295533/unit_circle_triangle_problem