To find the angle between the planes $AVB$ and $ABCD$ in this square-based pyramid, we can follow these steps:

1. Identify Key Components:

   • The square-based pyramid has a square base $ABCD$ and a vertex $V$ above the center of the base.
   • Plane $ABCD$ is the base plane of the pyramid, and $AVB$ is a triangular plane that includes edge $AV$ and point $B$.

2. Define the Angle Between the Planes:

   • The angle between two planes can be found by calculating the angle between their normal vectors.
   • However, in this problem, it is simpler to find the angle between line segment $VE$ (the altitude of triangle $AVB$) and line segment $AE$ (the base radius of the square). This angle will be the same as the angle between the planes.

3. Determine Triangle Measurements:

   • The pyramid has height $VE$ from the center $E$ of the square base to vertex $V$.
   • Given the side length $AD = 10$ cm, the length from the center $E$ to any vertex $A, B, C,$ or $D$ is $\frac{10}{2} = 5$ cm (since $E$ is the center of the square base).
   • $VA = 11.7$ cm is given.

4. Calculate $VE$ (Altitude of the Pyramid):

   • Using the Pythagorean theorem in right triangle $VAE$: $VE = \sqrt{VA^2 - AE^2} = \sqrt{11.7^2 - 5^2}$ $VE = \sqrt{136.89 - 25} = \sqrt{111.89} \approx 10.58 \text{ cm}$

5. Find the Angle $\angle AVE$:

   • Use trigonometry (cosine of angle $\theta$): $\cos(\angle AVE) = \frac{AE}{VA} = \frac{5}{11.7}$ $\cos(\angle AVE) \approx 0.4274$ $\angle AVE \approx \cos^{-1}(0.4274) \approx 64^\circ$

Answer:

The angle between the planes $AVB$ and $ABCD$ is approximately $64^\circ$.

Would you like more details on any step? Here are some related questions for practice:

1. How would the angle change if $VA$ were longer?
2. What if the base of the pyramid was a different shape, like a pentagon?
3. Can we find the angle using vector methods for normal vectors?
4. How does the height of the pyramid impact this angle?
5. What trigonometric identities help in finding angles in 3D shapes?

Tip: When working with angles in 3D geometry, visualize or draw perpendicular heights from key points to help define right triangles.