1. **State the problem:** We have a square ABCD with side length 10 cm. Two arcs are drawn inside the square: one connecting points A and B, and another connecting points C and D. These arcs intersect inside the square, creating a shaded "bowtie" shape in the lower left and right corners. We need to find the area of this shaded region in terms of $\pi$. 2. **Understand the arcs:** Each arc is part of a circle with radius equal to the side length of the square, 10 cm, centered at the opposite corner. For example, the arc from A to B is part of a circle centered at D, and the arc from C to D is part of a circle centered at B. 3. **Calculate the area of one circular segment:** The arcs form circular segments inside the square. The area of a circular segment is given by: $$\text{Segment area} = r^2 \arccos\left(\frac{d}{r}\right) - d \sqrt{r^2 - d^2}$$ where $r$ is the radius and $d$ is the distance from the center to the chord. 4. **Apply to our problem:** Here, the chord length is the side of the square, 10 cm, and the radius $r=10$ cm. The distance $d$ from the center to the chord is half the side length, $d=5$ cm. 5. **Calculate the segment area:** $$\arccos\left(\frac{5}{10}\right) = \arccos(0.5) = \frac{\pi}{3}$$ $$\sqrt{10^2 - 5^2} = \sqrt{100 - 25} = \sqrt{75} = 5\sqrt{3}$$ So, $$\text{Segment area} = 10^2 \times \frac{\pi}{3} - 5 \times 5\sqrt{3} = 100 \times \frac{\pi}{3} - 25\sqrt{3} = \frac{100\pi}{3} - 25\sqrt{3}$$ 6. **Calculate the area of the lens (bowtie) shape:** The shaded region is the intersection of two such segments, which is twice the area of one segment minus the square area. However, since the arcs intersect symmetrically, the shaded area is twice the area of one segment minus the square area: $$\text{Shaded area} = 2 \times \text{Segment area} - 100 = 2 \left(\frac{100\pi}{3} - 25\sqrt{3}\right) - 100 = \frac{200\pi}{3} - 50\sqrt{3} - 100$$ 7. **Express the shaded area in terms of $\pi$ only:** The problem asks for the area in terms of $\pi$, so we keep the $\pi$ term and note the constants. **Final answer:** $$\boxed{\text{Shaded area} = \frac{200\pi}{3} - 50\sqrt{3} - 100 \text{ cm}^2}$$ This is the exact area of the shaded bowtie region inside the square in terms of $\pi$ and radicals.