The area of a circle inscribed in a square is 7cm ^ 2, find the area of a circle described around this square.

Let a square ABCD be given with side AB = a. It is known that a circle is inscribed in a square with an area of ​​Svp = 7 cm ^ 2. The area of ​​the circle is found by the formula: Svp = π ∙ r ^ 2, but the diameter of the inscribed circle is equal to the length of the side of the square, then the radius r = a: 2, or Svp = π ∙ (a: 2) ^ 2 = (π ∙ a ^ 2 )/four. Let us express from this formula the length of the side of the square: a = 2 (Svn / π) ^ (1/2). To find the area of ​​the circumscribed circle, you need to find its radius R, which is half the length of the diagonal of the square. Let us express the length of the diagonal of the square d through the lengths of its sides, using the Pythagorean theorem: d ^ 2 = a ^ 2 + a ^ 2 = 2 ∙ a ^ 2; d = a ∙ 2 ^ (1/2); d = 2 ∙ (Svp / π) ^ (1/2) ∙ 2 ^ (1/2) = 2 ∙ (2 ∙ Svp / π) ^ (1/2). The radius of the circumscribed circle will be R = (2 ∙ Svp / π) ^ (1/2), and the area Sop = π ∙ R ^ 2 = π ∙ (2 ∙ S / π) ^ (1/2) ^ 2 = π ∙ (2 ∙ S / π) = 2 ∙ Svp = 2 ∙ 7 = 14 (cm ^ 2).
Answer: 14 sq. cm – the area of ​​a circle described around this square.