4. The surface area (in m²) of the largest sphere that can fit into a hollow cube with edges of length 1 meter is __________.
Given data: π = 3.14
Largest Sphere Surface Area in 1m Cube Explained
The largest sphere fitting inside a hollow cube of 1-meter edge length touches all six faces, making its diameter equal to the cube’s edge. With π = 3.14, the surface area calculates to 3.14 m².
Correct Calculation Step-by-Step
The sphere’s diameter matches the cube’s 1 m edge, so radius r = 0.5 m. Surface area formula is 4πr²: 4 × 3.14 × (0.5)² = 4 × 3.14 × 0.25 = 3.14 m². This fits perfectly as the inscribed sphere.
Common Options and Explanations
Multiple-choice traps often confuse sphere-cube relations; here’s a breakdown:
| Option | Value (m²) | Why Incorrect/Correct | Explanation |
|---|---|---|---|
| A | 3.14 | Correct | 4π(0.5)² with π=3.14 yields exactly 3.14; diameter = edge length. |
| B | 12.56 | Incorrect | Uses full 1 m radius: 4π(1)² = 12.56; too large for cube. |
| C | 4.71 | Incorrect | Wrong formula or diagonal (ignores face-touching fit). |
| D | 0.785 | Incorrect | Volume mistaken for area or r=0.25. |
Geometry Visualization
Imagine the cube’s interior: sphere centers at cube center, expands until contacting faces. Diameter constraint ensures maximal size without overlap. For cube-in-sphere (reverse), diameter uses space diagonal √3 m, but query specifies sphere-in-cube.
