The diameter of circle = 10cm
Radius = 5cm
Notice that the diagonal of the shaded square is actually a radius of a circle (it starts from the centre of the circle and ends at the circumference)
The two diameters intersect/meet at the circle's centre. So the bottom left corner of the square is the centre of the circle.
Now if we draw both diagonals of the square, the square is divided into 4 identical right-angled isosceles triangles. Each diagonal is divided into two by the other diagonal, so each section is 5cm÷ 2 = 2.5cm.
This also means the perpendicular sides of the triangles are 2.5cm long.
Area of square = area of the 4 triangles
= 4 × ½ × 2.5cm × 2.5cm
= 2 × 2.5cm × 2.5cm (basically the 4 triangles can be rearranged into two 2.5cm by 2.5cm squares)
= 2 × 6.25cm²
= 12.5cm²
Further info: the area of a square also equals ½d² (or ½ × d × d), where d is the length of its diagonal).