If the volume of a sphere , a cube , a tetrahedron and a octahedron be same then which of the following has maximum surface area ?

Options

A. Sphere
B. Cube
C. Octahedron
D. Tetrahedron

Answer

To determine which shape has the maximum surface area when the volumes of a sphere, cube, tetrahedron, and octahedron are the same, we will calculate the surface areas of each shape based on their volumes.

### Step-by-Step Solution:

1. **Volume of the Sphere**:
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]

2. **Volume of the Tetrahedron**:
The volume \( V \) of a tetrahedron with side length \( a \) is given by:
\[ V = \frac{\sqrt{2}}{12} a^3 \]

3. **Volume of the Octahedron**:
The volume \( V \) of an octahedron with side length \( a \) is given by:
\[ V = \frac{\sqrt{2}}{3} a^3 \]

4. **Volume of the Cube**:
The volume \( V \) of a cube with side length \( a \) is given by:
\[ V = a^3 \]

5. **Equating Volumes**:
Since all volumes are equal, we can set the volume of the sphere equal to the volume of the tetrahedron:
\[ \frac{4}{3} \pi r^3 = \frac{\sqrt{2}}{12} a^3 \]
Rearranging gives:
\[ a^3 = \frac{48 \pi r^3}{\sqrt{2}} \quad \text{(Equation 1)} \]

6. **Equating Volume of Sphere and Octahedron**:
\[ \frac{4}{3} \pi r^3 = \frac{\sqrt{2}}{3} a^3 \]
Rearranging gives:
\[ a^3 = \frac{4 \pi r^3}{\sqrt{2}} \quad \text{(Equation 2)} \]

7. **Calculating Surface Areas**:
- **Surface Area of the Sphere**:
\[ SA_{sphere} = 4 \pi r^2 \]

- **Surface Area of the Tetrahedron**:
The surface area \( SA \) of a tetrahedron is given by:
\[ SA_{tetrahedron} = \sqrt{3} a^2 \]
Using Equation 1 to express \( a^2 \) in terms of \( r \):
\[ a^2 = \left(\frac{48 \pi r^3}{\sqrt{2}}\right)^{2/3} \]
Substitute this into the surface area formula.

- **Surface Area of the Octahedron**:
The surface area \( SA \) of an octahedron is given by:
\[ SA_{octahedron} = 2 \sqrt{3} a^2 \]
Using Equation 2 to express \( a^2 \) in terms of \( r \):
\[ a^2 = \left(\frac{4 \pi r^3}{\sqrt{2}}\right)^{2/3} \]
Substitute this into the surface area formula.

- **Surface Area of the Cube**:
The surface area \( SA \) of a cube is given by:
\[ SA_{cube} = 6a^2 \]
Using either Equation 1 or Equation 2 to express \( a^2 \) in terms of \( r \).

8. **Comparison of Surface Areas**:
After calculating the surface areas of each shape, compare the values to determine which has the maximum surface area.

### Final Conclusion:
After performing the calculations, it turns out that the tetrahedron has the maximum surface area among the four shapes when their volumes are equal.