To solve the problem, we start by denoting the side length of the square as (a) and its area as (S = a^2). The shaded region (central overlapping area of the four quarter-circles) is known to be ((2 - \sqrt{3})a^2).

Key Observations:

• The shaded area is given as (10\ \text{cm}^2).
• The shaded area formula: ((2 - \sqrt{3})a^2 = 10).

Calculation:

[a^2 = \frac{10}{2 - \sqrt{3}}] Rationalize the denominator: [a^2 = \frac{10(2 + \sqrt{3})}{(2 - \sqrt{3})(2 + \sqrt{3})} = \frac{10(2 + \sqrt{3})}{4 - 3} = 10(2 + \sqrt{3})]

Answer: (\boxed{10(2 + \sqrt{3})}) (or (\boxed{20 + 10\sqrt{3}}) if expanded)

(\boxed{20 + 10\sqrt{3}})

作者声明:本文包含人工智能生成内容。