1. **State the problem:** We have a figure made from intersecting semicircles each with diameter $\frac{1}{2}$. We need to find the exact area of the shaded portion formed by these semicircles inside a square. 2. **Identify key dimensions:** The diameter of each semicircle is $\frac{1}{2}$, so the radius $r = \frac{1}{4}$. 3. **Understand the figure:** The square contains four leaf-like shapes formed by arcs of these semicircles. Each leaf is created by the intersection of two semicircles. 4. **Calculate the area of one leaf:** Each leaf is the intersection of two semicircles of radius $\frac{1}{4}$. The area of one semicircle is $$\frac{1}{2} \pi r^2 = \frac{1}{2} \pi \left(\frac{1}{4}\right)^2 = \frac{1}{2} \pi \frac{1}{16} = \frac{\pi}{32}.$$ 5. **Area of intersection (leaf):** The leaf shape is the lens formed by two semicircles intersecting at right angles. The area of the lens formed by two circles of radius $r$ intersecting with centers a distance $r$ apart is known to be $$A_{leaf} = r^2 \left(\pi - 2\right).$$ 6. **Substitute $r=\frac{1}{4}$:** $$A_{leaf} = \left(\frac{1}{4}\right)^2 (\pi - 2) = \frac{1}{16} (\pi - 2) = \frac{\pi - 2}{16}.$$ 7. **Total shaded area:** There are 4 such leaves inside the square, so total shaded area is $$4 \times \frac{\pi - 2}{16} = \frac{4(\pi - 2)}{16} = \frac{\pi - 2}{4}.$$ **Final answer:** $$\boxed{\frac{\pi - 2}{4}}.$$