show the formula or a way step by step instead of not showing it.

Sure! I'll walk you through the step-by-step calculation to find the angle between the diagonal of the cuboid and the length of the cuboid.

### Step 1: Understand the Geometry

The cuboid has dimensions:
- Length \( = 16.2 \, \text{cm} \)
- Width \( = 5.5 \, \text{cm} \)
- Height \( = 8.0 \, \text{cm} \)

The diagonal \( AB \) runs from one corner of the cuboid to the opposite corner. We are calculating the angle \( \theta \) between this diagonal and one of the edges of the cuboid (specifically, the length \( 16.2 \, \text{cm} \)).

### Step 2: Calculate the Length of the Space Diagonal

The length of the space diagonal \( AB \) can be found using the 3D Pythagorean Theorem:
\[
AB = \sqrt{( \text{Length})^2 + (\text{Width})^2 + (\text{Height})^2}
\]

Substituting the values:
\[
AB = \sqrt{(16.2)^2 + (5.5)^2 + (8.0)^2}
\]
This gives:
\[
AB = \sqrt{262.44 + 30.25 + 64} = \sqrt{356.69} \approx 18.89 \, \text{cm}
\]

### Step 3: Use the Cosine Formula for the Angle

To find the angle \( \theta \) between the space diagonal \( AB \) and the edge (length of the cuboid), we use the cosine of the angle:
\[
\cos(\theta) = \frac{\text{Adjacent side (length)}}{\text{Hypotenuse (diagonal)}}
\]

Here, the adjacent side is the length of the cuboid \( 16.2 \, \text{cm} \) and the hypotenuse is the space diagonal \( AB \).

\[
\cos(\theta) = \frac{16.2}{18.89}
\]

Next, calculate the angle \( \theta \) by taking the inverse cosine (arccos):
\[
\theta = \arccos\left( \frac{16.2}{18.89} \right)
\]

### Step 4: Convert to Degrees

Performing the calculation:
\[
\theta \approx \arccos(0.857) \approx 30.93^\circ
\]

Thus, the angle \( \theta \) between the diagonal and the length of the cuboid is approximately \( 30.93^\circ \).

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This step-by-step breakdown shows how the formula is derived and calculated. Do you have any other specific questions on this, or would you like to explore further?

### Related Questions:
1. How would you apply a similar method to calculate the angle between the diagonal and the width?
2. What is the angle between the diagonal and the height of the cuboid?
3. How is the space diagonal affected by scaling the dimensions of the cuboid?
4. What is the relationship between the face diagonal and the space diagonal?
5. How would the formula change if we considered a cube instead of a cuboid?

### Tip:
When solving 3D geometry problems, always visualize the shape and clearly label each side, diagonal, and angle you're working with to keep the calculations organized and accurate.

Learn how to calculate the angle between the diagonal and the length of a cuboid step-by-step with a detailed mathematical solution. This problem involves using the 3D Pythagorean Theorem and the Cosine Rule. Suitable for advanced high school students and beyond.

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