https://r2.erweima.ai/mathbot/image/1719477625949h921uwfa.png

The problem in the image translates to:

"Given a cube with all its vertices on a sphere, if the surface area of the cube is 18, find the volume of the sphere."

Let's solve this step by step.

1. **Find the side length of the cube**:
   The surface area of a cube is given by \( 6a^2 \), where \( a \) is the side length of the cube. Given that the surface area is 18:
   \[
   6a^2 = 18 \implies a^2 = 3 \implies a = \sqrt{3}
   \]

2. **Find the space diagonal of the cube**:
   The space diagonal of a cube (which is the diameter of the sphere) is given by \( a\sqrt{3} \). So, the diameter \( d \) of the sphere is:
   \[
   d = \sqrt{3} \times \sqrt{3} = 3
   \]
   Hence, the radius \( r \) of the sphere is:
   \[
   r = \frac{d}{2} = \frac{3}{2} = 1.5
   \]

3. **Find the volume of the sphere**:
   The volume \( V \) of a sphere is given by \( \frac{4}{3} \pi r^3 \). Using \( r = 1.5 \):
   \[
   V = \frac{4}{3} \pi (1.5)^3 = \frac{4}{3} \pi \times 3.375 = \frac{4 \times 3.375}{3} \pi = 4.5 \pi
   \]

So, the volume of the sphere is \( 4.5 \pi \) cubic units.

Would you like more details or have any questions?

Here are some related questions you might find interesting:
1. What is the surface area of a sphere given its radius?
2. How do you derive the formula for the volume of a sphere?
3. How is the space diagonal of a cube related to its side length?
4. What are some real-world applications of these geometric concepts?
5. Can you solve similar problems involving other polyhedra inscribed in spheres?

**Tip:** When solving geometric problems involving spheres and cubes, always remember to clearly identify and use the relationships between the dimensions of the shapes involved.

Learn how to find the volume of a sphere inscribed in a cube with a surface area of 18 units. Step-by-step solution using geometric formulas for cube surface area and sphere volume. Suitable for high school students studying geometry.

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