Worked Solution: Volume of a Sphere with Diameter 2
When given the diameter of a sphere instead of the radius, the most critical first step is to halve the diameter to obtain the radius: r = d / 2 = 2 / 2 = 1.
⚡ Why Dividing by 2 First is Crucial:
Cubing the diameter directly ($2^3$) without dividing by 2 will result in an answer exactly 8 times too large (\(2^3 = 8\))! Alternatively, use the direct diameter formula: V = ¹⁄₆πd³.
Step-by-Step Calculation
| Step | Action | Formula & Working | Result |
|---|---|---|---|
| Step 1 | Calculate radius from diameter | r = d ÷ 2 = 2 ÷ 2 | 1 |
| Step 2 | Cube the radius | (1)³ = 1 × 1 × 1 | 1 |
| Step 3 | Apply V = ⁴⁄₃ × π × r³ | ⁴⁄₃ × 3.14159 × 1 | 4.19 |
| Alternative | Direct diameter formula: V = ¹⁄₆πd³ | ¹⁄₆ × 3.14159 × 8 | 4.19 |