How long would the sides of an equilateral triangle be in order to contain a circle with a radius of 15"?

Question

the triangle is circumscribed by the circle

Answer 1

Draw an equilateral triangle inscribed inside a circle.  Call the points of the triangle A, B, and C.  Let D be the center of the circle.

Notice that DA and DB are radii, so they're each 15".  Since the points of the triangle are equidistant, the angle ADB is 360/3 = 120.  We want to find AB, which is the length of one of the sides.

By the law of cosines:

AB^2 = DA^2 + DB^2 - 2(DA)(DB)cos(120)

AB^2 = 15^2 + 15^2 - 2(15)(15)cos(120)

AB^2 = 2(15^2) - 2(15^2)(-1/2)

AB^2 = 2(15^2) + (15^2)

AB^2 = 3(15^2)

AB = 15√3