A sphere passes through the eight corners of a cube of side 10cm find the volume of the sphere.

To find the volume of the sphere that passes through the eight corners of a cube with a side length of 10 cm, we can follow these steps:

### Step 1: Determine the diagonal of the cube
The diagonal \(d\) of a cube can be calculated using the formula:
\[
d = \sqrt{3} \times \text{side length}
\]
For a cube with a side length of 10 cm:
\[
d = \sqrt{3} \times 10 \text{ cm} \approx 17.32 \text{ cm}
\]

### Step 2: Relate the diagonal to the diameter of the sphere
Since the sphere passes through all eight corners of the cube, the diagonal of the cube is equal to the diameter of the sphere. Therefore, the diameter \(D\) of the sphere is 17.32 cm.

### Step 3: Calculate the radius of the sphere
The radius \(r\) of the sphere is half of the diameter:
\[
r = \frac{D}{2} = \frac{17.32}{2} \text{ cm} = 8.66 \text{ cm}
\]

### Step 4: Calculate the volume of the sphere
The volume \(V\) of a sphere is given by the formula:
\[
V = \frac{4}{3} \pi r^3
\]
Substituting the radius \(r = 8.66 \text{ cm}\):
\[
V = \frac{4}{3} \pi (8.66)^3 \text{ cm}^3 \approx 2722.54 \text{ cm}^3
\]

So, the volume of the sphere is approximately **2722.54 cm³**.

Would you like more details or have any questions? 

Here are five related questions to consider:
1. How is the diagonal of a cube related to its side length?
2. What would be the volume of a sphere that fits perfectly inside a cube with side length 10 cm?
3. How would the volume change if the side length of the cube doubled?
4. What is the surface area of the sphere in this problem?
5. Can the same approach be used if the shape is a rectangular prism instead of a cube?

**Tip:** Understanding the relationship between the geometry of a cube and a sphere is essential in solving problems involving both shapes.

Discover how to calculate the volume of a sphere passing through the eight corners of a cube with a side length of 10 cm. Includes step-by-step solution, formulas for sphere volume and cube diagonal, and related questions. Suitable for high school students.

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