# A regular triangle is inscribed in the circle, the side of which is 6 cm. Calculate the area of a square inscribed

**A regular triangle is inscribed in the circle, the side of which is 6 cm. Calculate the area of a square inscribed in the same circle.**

Since triangle ABC is regular, its internal angles are 60.

The inscribed angle ACB rests on the arc AB, then the degree measure of the arc AB = 120.

Then the central angle AOB is also based on the arc AB, and then the angle AOB = 120.

In an isosceles triangle AOB AO = OB = R, then, by the cosine theorem:

AB ^ 2 = R ^ 2 + R ^ 2 – 2 * R * R * Cos120.

36 = 2 * R ^ 2 – 2 * R ^ 2 * (-1/2).

3 * R ^ 2 = 36.

R ^ 2 = 12.

R = 2 * √3 cm.

The segment OM = R = 2 * √3 cm and is equal to half the diagonal of the square of the CMHR. Then MР = 4 * √3 cm.

Let’s define the area of a square in terms of its diagonal.

Skv = MР ^ 2/2 = 48/2 = 24 cm2.

Answer: The area of the square is 24 cm2.