The radius of the circle inscribed in the square is 5 cm. Find the diagonal of the square.

The sides AB and AD outgoing from the vertex A are two tangents to the inscribed circle. The radii OG and OI (perpendicular to the sides at the tangency points) and the segments AG and AI form a square with a side equal to the radius of the circle r. Similar small squares can be built on other vertices of the square.

Square ABCD consists of four squares with side r and area r ^ 2. Area of ​​a compound square in radii:

S = 4r ^ 2.

Side of a large square in radii:

a = √S = √ (4r ^ 2) = 2r.

Diagonal d of square ABCD:

AC = √AD ^ 2 + DC ^ 2 =
= √ ((2r) ^ 2 + (2r) ^ 2) =
= √ (8r ^ 2) = 2√2 * r =
= 2√2 * 5 cm = 10√2 cm ≈ 14.14 cm.

Answer: 10√2 cm or 14.14 cm.