Equilateral triangle ABC inscribed in a circle with a radius of 5 cm. Find the area and side.

Let us denote by a the length of the side of this equilateral triangle.

Since any angle of any equilateral triangle is 60 °, then applying the formula for the area of ​​a triangle on two sides and the angle between them, we find the area S of this triangle:

S = a * a * sin (60 °) / 2 = a ^ 2 * (√3 / 2) * 1/2 = a ^ 2 * √3 / 4 cm ^ 2.

Using the formula for the area of ​​a triangle in terms of the radius of the circumscribed circle, we can compose the following equation:

a ^ 2 * √3 / 4 = a ^ 3 / (4 * 5),

solving which, we get:

a ^ 2 * √3 / 4 = a ^ 3/20;

√3 / 4 = a / 20;

a = 20 * √3 / 4 = 5√3 cm.

Knowing the length of the side, we find the area S of this triangle:

a ^ 2 * √3 / 4 = (5√3) ^ 2 * √3 / 4 = 75√3 / 4 cm ^ 2.

Answer: the length of the side is 5√3 cm, the area of ​​the triangle is 75√3 / 4 cm ^ 2.