The area of an equilateral triangle is 3√3 cm2. Find the radius of a circle inscribed in a triangle.

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

According to the condition of the problem, the area of ​​this triangle is 3√3 cm ^ 2.

Since each angle of an equilateral triangle is 60 °, applying the formula for the area of ​​a triangle along two sides and the angle between them, we can compose the following equation:

a * a * sin (60 °) / 2 = 3√3,

solving which, we get:

a ^ 2 * √3 / 2 = 3√3;

a ^ 2 = 3√3 / (√3 / 2);

a ^ 2 = 6;

a = √6 cm.

Applying the formula for the area of ​​a triangle in terms of the radius r of the inscribed circle, we find r:

r = 2 * 3√3 / (√6 + √6 + √6) = 2 * 3√3 / (3√6) = 2 / √2 = √2 cm.

Answer: the radius of the inscribed circle is √2 cm.