Shaded Area Circle Square 0Aa277

1. **Problem Statement:** We have a square of side length $14$ cm containing a circle centered at $O$ touching all four sides, and two quarter circles drawn from the lower left and lower right corners. We need to find the total area of the shaded parts. 2. **Identify the shapes and their dimensions:** - Square side length $= 14$ cm. - Circle inside the square touches all sides, so its diameter $= 14$ cm, radius $r = \frac{14}{2} = 7$ cm. - Two quarter circles each have radius $= 7$ cm (since they are drawn from corners to bottom center). 3. **Calculate areas:** - Area of square: $$14 \times 14 = 196 \text{ cm}^2$$ - Area of circle: $$\pi r^2 = \frac{22}{7} \times 7^2 = \frac{22}{7} \times 49 = 154 \text{ cm}^2$$ - Area of one quarter circle: $$\frac{1}{4} \pi r^2 = \frac{1}{4} \times 154 = 38.5 \text{ cm}^2$$ - Area of two quarter circles: $$2 \times 38.5 = 77 \text{ cm}^2$$ 4. **Analyze shaded area:** - The shaded area is the area of the circle plus the two quarter circles minus the overlapping white leaf-shaped regions formed by the quarter circles inside the square. - The two quarter circles overlap inside the circle, creating two white leaf-shaped regions. - The total shaded area is the area of the circle plus the two quarter circles minus the overlapping white regions. 5. **Calculate the overlapping white leaf-shaped regions:** - The two quarter circles together form a semicircle of radius $7$ cm at the bottom. - The circle inside the square is also of radius $7$ cm. - The overlapping white region is the area of the semicircle formed by the two quarter circles minus the area of the bottom semicircle of the circle. - Area of semicircle from quarter circles: $$\frac{1}{2} \pi r^2 = \frac{1}{2} \times 154 = 77 \text{ cm}^2$$ - Area of bottom semicircle of the circle: $$\frac{1}{2} \times 154 = 77 \text{ cm}^2$$ - Since these two semicircles coincide, the overlapping white region is zero. 6. **Therefore, total shaded area:** - Shaded area = Area of square $-$ (area of two quarter circles + area of circle) + overlapping white regions - But since the quarter circles and circle overlap inside the square, the shaded area is the area of the square minus the white leaf-shaped regions. 7. **Calculate white leaf-shaped regions:** - White leaf-shaped regions = area of two quarter circles $-$ area of bottom semicircle of circle - Both are equal to $77$ cm², so white leaf-shaped regions = $0$ cm². 8. **Hence, total shaded area = area of circle + area of two quarter circles $-$ white leaf-shaped regions = 154 + 77 - 0 = 231$ cm².** **Final answer:** $$\boxed{231 \text{ cm}^2}$$