An equilateral triangle is inscribed in a circle of radius r. Express the circumference C of the circle as a function of the length x of a side of the triangle

Question

An equilateral triangle is inscribed in a circle of radius r. Express the circumference C of the circle as a function of the length x of a side of the triangle.

Answer 1

Draw an equilateral triangle inscribed in a circle with radius r.

From the center draw the radius r to a vertex of the triangle

From the center draw the perpendicular to a side of the triangle x.

Two 30-60-90 triangles are created. 

The long side of each triangle is (r√3)/2.

Two long sides = x. x = r√3

r = x/√3

r = (x√3)/3

C = 2π((x√3)/3)

C = 2πx√3 / 3

 

Simplify according to instructions.