Find the radii of the circles inscribed in and circumscribed about a regular triangle if the side of the triangle is 4 cm.

Let’s draw medians in a triangle (They are bisectors, they are mediatrists). Median / side = COS (30) (according to the drawing) => Median / 4 = √ (3) / 3 => Median = 4 * √ (3) / 3 We denote one median AB, the intersection point of the medians – O. The center of the inscribed and the center of the circumscribed circle is located at point O. AB – median => AO / 2 = OB / 1 = AB / 3 => AO (Radius of the circumcircle) = AB * 2/3 => AO = 8 * √ (3) / 9 OB (Inscribed circle radius) => AB / 3 => OB = 4 * √ (3) / 9