A triangle with 2 sides equal in R√3 is inscribed in a circle of radius R. Find the area of the triangle.

Let’s build the radii OA, OB, OC.

The triangle AOB is isosceles, in which we apply the cosine theorem and determine the cosine of the angle AOB.

AB ^ 2 = AO ^ 2 + BO ^ 2 – 2 * AO * BO * CosAOB.

3 * R ^ 2 = R ^ 2 + R ^ 2 – 2 * R * R * CosAOB.

2 * R ^ 2 * CosAOB = – R ^ 2.

CosAOB = -1/2.

Angle AOB = 120.

Then the angle BOC = 120, since the triangles AOB and BOC are equal on three sides.

Then the angle AOC = 360 – 120 – 120 = 120, and therefore, the triangle ABC is equilateral with a side R * √3 cm, since the vertices of the triangle divide the circle into equal arcs of 120.

Then Savs = AB ^ 2 * √3 / 4 = R ^ 2 * 3 * √3 / 4 cm2.

Answer: The area of the triangle is R2 * 3 * √3 / 4 cm2.