A regular triangle is inscribed in a circle and a regular triangle is described around the circle. Find: the ratio of their areas.

Let the side of a triangle inscribed in a circle be equal to a, then its area is expressed by the formula:

S1 = a³ / 4 * R, where R is the radius of the circle.

The area of a triangle circumscribed around a circle is expressed by the formula:

S2 = 3 * b * R / 2, where b is the side of the triangle and R is the radius of the inscribed circle.

Let’s calculate the ratio of the areas:

S2 / S1 = (3 * b * R / 2) * (4 * R / a³) = 6 * b * R² / a³.

Since the first triangle is regular, we write its side through the formula:

a = R * √3.

Then S2 / S1 = (6 * b * R²) / (R * √3) ³;

S2 / S1 = 2 * b / √3 * R.

Answer: S2 / S1 = 2 * b / √3 * R.