The volume of the largest sphere which can be carved out of a cube of side $6$ cm is

### Concept & Inscribed Sphere The largest sphere that can be carved from a cube perfectly touches the centers of the cube's six faces. Therefore, the diameter of this largest sphere is exactly equal to the side length of the cube. $$d_{\text{sphere}} = a_{\text{cube}}$$ Volume of a sphere $= \frac{4}{3}\pi r^3$. ### Step-by-Step Solution * **Given:** Side of the cube $a = 6$ cm. * Diameter of the largest possible sphere $d = 6$ cm. * Radius of the sphere $r = \frac{6}{2} = 3$ cm. * Volume of the sphere $= \frac{4}{3} \times \pi \times r^3$. * Volume $= \frac{4}{3} \times \frac{22}{7} \times 3^3 = \frac{4}{3} \times \frac{22}{7} \times 27$. * Volume $= 4 \times \frac{22}{7} \times 9 = \frac{792}{7}$. * Convert to decimal: $792 \div 7 = 113.1428... \text{ cm}^3$. ### Exam Strategy & Shortcut Recognize that the volume of a sphere inscribed in a cube is about $52.4\%$ of the cube's volume ($\frac{\pi}{6}$). The cube's volume is $6^3 = 216$. Roughly half of $216$ is $108$. The only option close to this estimation is $113.14$. ### Common Pitfall Using the cube's diagonal to find the sphere's radius. The diagonal determines the *circumscribed* sphere (outside the cube), but a *carved* sphere is inscribed (inside), limited by the flat faces, not the corners. ### Final Answer Therefore, the correct answer is **$113.14 \text{ cm}^3$**.