A square with a side of 7√2 cm is inscribed in a circle. Find the side of a regular triangle in which the same circle is inscribed.

Let us find out what the diagonal of this square will equal, when it is precisely known from the condition of our problem that the length of the side is equal to 7√2 cm:

√ (2 * (7√2) ^ 2) = 14.

We will find out what, in this case, the radius of the circle in which it is inscribed will correspond, because we know that it is equal to half of the diagonal found above:

14: 2 = 7.

We find out what value will correspond to the length of the side of a regular triangle, in which, according to the condition, a circle is inscribed, with a radius that was found above:

r = a / 2√3;

a = 7 * 2√3 = 14√3.

Answer: 14√3 cm.