The length of a circle inscribed in a regular triangle is 2 √3п cm. Find the length of a circle circumscribed about this triangle.

The length of a circle inscribed in a triangle:

l = 2 п r, where r is the radius of the inscribed circle;

r = l: 2 п= 2√3 п: 2 п = √3 (cm);

The radius of a circle inscribed in a regular triangle:

r = a / (2 * √3), and is the side of the triangle;

a = r * √3 = 2 √3 * √3 = 6 (cm);

Radius of the circumscribed circle:

R = a / √3 = 6: √3 = 2 √3 (cm);

The length of a circle around a triangle:

L = 2 R = 2 * п * 2 √3 = 4 п √3 (cm).