The triangle KMN has an angle K = 80, an angle N = 40, and the KN side has a length of 6cm.

The triangle KMN has an angle K = 80, an angle N = 40, and the KN side has a length of 6cm. Find the radius of the circle around the triangle.

It is known from the extended sine theorem that:
k / sinK = m / sinM = n / sinN = 2R,
where k, m and n are the sides of an arbitrary triangle opposite to the angles K, M and N, respectively. In the triangle KMN opposite the angle K lies the side MN, opposite the angle M lies the side KN, opposite the angle N lies the side KM, then:
MN / sinK = KN / sinM = KM / sinN = 2R.
Find the degree measure of the angle M. By the theorem on the sum of the angles of a triangle:
angle K + angle M + angle N = 180 degrees;
80 + angle M + 40 = 180;
angle M = 180 – 120;
angle M = 60 degrees.
Hence:
KN / sinM = 2R;
6 / sin60 = 2R;
6 / √3 / 2 = 2R;
12 / √3 = 2R;
2√3R = 12 (proportional);
R = 12 / 2√3;
R = 6 / √3 = 6√3 / 3 = 2√3 (cm).
Answer: R = 2√3 cm.