If the area of three adjacent faces of a cuboid are X, Y and Z respectively, then find the volume of cuboid.

Answer

To find the volume of a cuboid given the areas of three adjacent faces, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Variables**:
Let the dimensions of the cuboid be:
- Length = \( L \)
- Breadth = \( B \)
- Height = \( H \)

2. **Relate the Areas to Dimensions**:
The areas of the three adjacent faces can be expressed as:
- Area of face with dimensions \( L \) and \( B \): \( A_1 = L \times B = X \)
- Area of face with dimensions \( B \) and \( H \): \( A_2 = B \times H = Y \)
- Area of face with dimensions \( H \) and \( L \): \( A_3 = H \times L = Z \)

3. **Express the Dimensions in Terms of Areas**:
From the equations above, we can express the dimensions in terms of the areas:
- From \( X = L \times B \), we can write \( L = \frac{X}{B} \)
- From \( Y = B \times H \), we can write \( H = \frac{Y}{B} \)
- From \( Z = H \times L \), we can substitute \( H \) and \( L \) into this equation.

4. **Substituting Dimensions**:
Substitute \( L \) and \( H \) into the equation for \( Z \):
\[ Z = H \times L = \left(\frac{Y}{B}\right) \times \left(\frac{X}{B}\right) = \frac{XY}{B^2} \]
Rearranging gives us:
\[ B^2 = \frac{XY}{Z} \quad \Rightarrow \quad B = \sqrt{\frac{XY}{Z}} \]

5. **Finding Length and Height**:
Now, substitute \( B \) back to find \( L \) and \( H \):
- For \( L \):
\[ L = \frac{X}{B} = \frac{X}{\sqrt{\frac{XY}{Z}}} = \frac{X \sqrt{Z}}{\sqrt{XY}} = \sqrt{\frac{X^2Z}{Y}} \]
- For \( H \):
\[ H = \frac{Y}{B} = \frac{Y}{\sqrt{\frac{XY}{Z}}} = \frac{Y \sqrt{Z}}{\sqrt{XY}} = \sqrt{\frac{Y^2Z}{X}} \]

6. **Volume of the Cuboid**:
The volume \( V \) of the cuboid is given by:
\[ V = L \times B \times H \]
Substituting the expressions for \( L \), \( B \), and \( H \):
\[ V = \left(\sqrt{\frac{X^2Z}{Y}}\right) \times \left(\sqrt{\frac{XY}{Z}}\right) \times \left(\sqrt{\frac{Y^2Z}{X}}\right) \]

7. **Simplifying the Volume Expression**:
Combining these expressions:
\[ V = \sqrt{\frac{X^2Z}{Y} \cdot \frac{XY}{Z} \cdot \frac{Y^2Z}{X}} = \sqrt{XYZ} \]

8. **Final Result**:
Therefore, the volume of the cuboid is:
\[ V = \sqrt{XYZ} \]