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The pyramid in the picture is regular. Calculate the measure of the angle of the side wall to the base.

To calculate the angle between the slanted wall of the pyramid and its base, we need to understand the geometry of the pyramid.

### Key Information:
- This is a regular pyramid, meaning its base is a regular polygon (here, a square) and the apex is directly above the center of the base.
- The side length of the base is \( a \).
- The slant height (the distance from the apex to the midpoint of a side of the base) is given as \( \frac{\sqrt{7}}{2} a \).

We are asked to calculate the angle \( \theta \) between the slant height (side wall) and the base.

### Solution Approach:

The pyramid forms a right triangle with:
- The slant height \( l = \frac{\sqrt{7}}{2} a \),
- Half the base side length \( \frac{a}{2} \),
- The perpendicular height from the apex to the center of the base, \( h \), which can be found using the Pythagorean theorem.

First, we find the height \( h \) of the pyramid using the relationship in the right triangle formed by \( h \), \( \frac{a}{2} \), and \( l \) (the slant height).

From the Pythagorean theorem:
\[
l^2 = h^2 + \left(\frac{a}{2}\right)^2
\]
\[
\left(\frac{\sqrt{7}}{2} a\right)^2 = h^2 + \left(\frac{a}{2}\right)^2
\]
Now, I will calculate the height \( h \) and then use trigonometry to find the angle \( \theta \).The height \( h \) of the pyramid is \( \frac{\sqrt{6}}{2} a \), as we take the positive root for physical reasons (heights cannot be negative).

Now, to find the angle \( \theta \) between the slant height and the base, we can use the tangent function in the right triangle formed by \( h \) (the vertical height) and half the base side length \( \frac{a}{2} \):

\[
\tan(\theta) = \frac{h}{\frac{a}{2}} = \frac{\frac{\sqrt{6}}{2} a}{\frac{a}{2}} = \sqrt{6}
\]

Next, we will calculate \( \theta \) as:

\[
\theta = \tan^{-1}(\sqrt{6})
\]The angle \( \theta \) between the slanted wall of the pyramid and the base is approximately \( 67.79^\circ \).

Would you like more details or have any questions about the solution?

Here are 5 related questions you might find helpful:
1. How would the angle change if the slant height was different?
2. How do you calculate the surface area of this pyramid?
3. What is the formula for the volume of a pyramid?
4. Can you calculate the angle between the height of the pyramid and the base?
5. What happens to the angle as the side length of the base increases?

**Tip**: In geometry, visualizing right triangles formed by key components (such as height and half the base) can simplify finding angles in 3D shapes like pyramids.

Learn how to calculate the angle between the slanted wall and the base of a regular pyramid. This problem covers key geometric and trigonometric concepts like the Pythagorean theorem and tangent function. Suitable for students in Grades 10-12.

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