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The Diagram shows a cuboid with dimensions 5.5cm, 8cm, and 16.2cm
Calculate the angle between the line AB

The cuboid in the image has dimensions \( 5.5 \, \text{cm} \), \( 8 \, \text{cm} \), and \( 16.2 \, \text{cm} \). You are asked to calculate the angle between the line \( AB \), which is the diagonal running from one vertex of the cuboid to the opposite vertex, and one of the edges of the cuboid.

Let's denote:
- \( A \) and \( B \) as the diagonal's endpoints.
- \( AB \) is the space diagonal of the cuboid.
- The dimensions are:
  - Length: \( 16.2 \, \text{cm} \)
  - Width: \( 5.5 \, \text{cm} \)
  - Height: \( 8 \, \text{cm} \)

### Approach

To calculate the angle between line \( AB \) (space diagonal) and one of the cuboid's edges, we'll:

1. **Find the length of the space diagonal** \( AB \) using the 3D version of the Pythagorean theorem:
   \[
   AB = \sqrt{(16.2)^2 + (5.5)^2 + (8)^2}
   \]

2. **Determine the angle** between \( AB \) and a specific edge (e.g., the length \( 16.2 \, \text{cm} \)) by applying trigonometric relationships.

The angle \( \theta \) between the space diagonal and the length can be calculated using:
\[
\cos(\theta) = \frac{\text{Length of the edge}}{\text{Length of the diagonal}}
\]

Let's compute this now.import math

# Dimensions of the cuboid
length = 16.2  # in cm
width = 5.5    # in cm
height = 8.0   # in cm

# Calculating the length of the space diagonal (AB) using Pythagoras theorem in 3D
diagonal_AB = math.sqrt(length**2 + width**2 + height**2)

# Calculating the angle between AB and the length of the cuboid (16.2 cm)
cos_theta = length / diagonal_AB
theta_rad = math.acos(cos_theta)

# Converting the angle from radians to degrees
theta_deg = math.degrees(theta_rad)

diagonal_AB, theta_degThe length of the space diagonal \( AB \) is approximately \( 18.89 \, \text{cm} \).

The angle \( \theta \) between the diagonal \( AB \) and the length of the cuboid (16.2 cm) is approximately \( 30.93^\circ \).

Would you like any further details, or do you have more questions on this topic?

### Related Questions:
1. How do you calculate the angle between the diagonal and a different edge, like the width?
2. What is the length of the diagonal of the face of the cuboid instead of the space diagonal?
3. How can we find the angle between the diagonal and the height of the cuboid?
4. What is the significance of the Pythagorean theorem in 3D space?
5. Can this method be applied to other 3D shapes, such as pyramids or spheres?

### Tip:
When solving problems involving angles in 3D space, breaking down the problem into right triangles can help simplify the calculations by allowing you to apply the Pythagorean theorem and trigonometry.

Learn how to calculate the angle between the space diagonal AB of a cuboid with dimensions 5.5 cm, 8 cm, and 16.2 cm. Detailed solution using the 3D Pythagorean theorem and trigonometry. Suitable for high school students studying geometry and trigonometry.

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