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To calculate the size of angle \( CAG \) in the cuboid, we need to analyze the dimensions given and understand the 3D positioning of the points \( A \), \( C \), and \( G \).

### Step 1: Identify the Position of Points
In this cuboid:
- \( A \) is at the origin point of the cuboid.
- \( B \) is along the width from \( A \) (27 cm away).
- \( C \) is along the length from \( B \) (36 cm away from \( A \) along one dimension).
- \( G \) is along the height of the cuboid from \( C \) (31.5 cm away from \( C \)).

We can consider:
- \( A \) at \( (0, 0, 0) \),
- \( C \) at \( (36, 0, 0) \),
- \( G \) at \( (36, 27, 31.5) \).

### Step 2: Use the Cosine Rule to Calculate \( \angle CAG \)
To find \( \angle CAG \), we can apply the cosine rule in 3D, where:
\[
\cos(\angle CAG) = \frac{\overrightarrow{CA} \cdot \overrightarrow{AG}}{|\overrightarrow{CA}| \cdot |\overrightarrow{AG}|}
\]
where:
- \( \overrightarrow{CA} = (36, 0, 0) - (0, 0, 0) = (36, 0, 0) \),
- \( \overrightarrow{AG} = (36, 27, 31.5) - (0, 0, 0) = (36, 27, 31.5) \).

#### Calculate \( \overrightarrow{CA} \cdot \overrightarrow{AG} \):
\[
\overrightarrow{CA} \cdot \overrightarrow{AG} = (36 \times 36) + (0 \times 27) + (0 \times 31.5) = 1296
\]

#### Calculate \( |\overrightarrow{CA}| \) and \( |\overrightarrow{AG}| \):
- \( |\overrightarrow{CA}| = \sqrt{36^2 + 0^2 + 0^2} = 36 \)
- \( |\overrightarrow{AG}| = \sqrt{36^2 + 27^2 + 31.5^2} = \sqrt{1296 + 729 + 992.25} = \sqrt{3017.25} \approx 54.93 \)

#### Calculate \( \cos(\angle CAG) \):
\[
\cos(\angle CAG) = \frac{1296}{36 \times 54.93} = \frac{1296}{1977.48} \approx 0.6555
\]

#### Calculate \( \angle CAG \):
\[
\angle CAG = \cos^{-1}(0.6555) \approx 49.1^\circ
\]

### Final Answer:
\[
\angle CAG \approx 49.1^\circ
\]

Would you like further details on the steps, or do you have any questions?

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Here are some related questions:
1. How would the calculation change if the dimensions of the cuboid were different?
2. Can we find other angles in this cuboid using similar methods?
3. What would be the diagonal distance between two opposite corners of the cuboid?
4. How can we find the volume and surface area of this cuboid?
5. What if we wanted to find the distance between points \( A \) and \( G \)?

**Tip:** When calculating angles in 3D shapes, it's often helpful to visualize or sketch the vectors involved to ensure accuracy.

Calculate the size of angle CAG in a cuboid with dimensions 27 cm, 36 cm, and 31.5 cm.

Learn how to calculate the size of angle CAG in a cuboid with dimensions 27 cm, 36 cm, and 31.5 cm. This step-by-step guide covers 3D geometry concepts, vector dot product, and the cosine rule. Suitable for Grades 10-12.

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