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The question asks us to find the area of the base of a regular square pyramid, given that its lateral faces are equilateral triangles and its altitude is 8 inches.

### Solution Steps:
1. **Understanding the Geometry**:
   - The base of the pyramid is a square.
   - The lateral faces are equilateral triangles.
   - The altitude (height from the apex to the base) is 8 inches.

2. **Relations in the Pyramid**:
   - Since the lateral faces are equilateral triangles, the slant height \( s \) of the pyramid (the height of the triangular faces) is equal to the side of the base square. Let the side of the base be \( a \).
   - The altitude of the pyramid is related to the slant height \( s \) and the altitude of the equilateral triangles.

3. **Pythagoras Theorem Application**:
   The height of the pyramid forms a right triangle with half the base \( \frac{a}{2} \) and the altitude. Using Pythagoras’ theorem:
   \[
   s^2 = \left(\frac{a}{2}\right)^2 + 8^2
   \]
   \[
   s^2 = \frac{a^2}{4} + 64
   \]

   Since the slant height \( s \) is the same as the side length of the square base \( a \) (due to the lateral faces being equilateral triangles):
   \[
   a^2 = \frac{a^2}{4} + 64
   \]
   Multiply through by 4 to eliminate the fraction:
   \[
   4a^2 = a^2 + 256
   \]
   \[
   3a^2 = 256
   \]
   \[
   a^2 = \frac{256}{3}
   \]
   \[
   a = \frac{16}{\sqrt{3}} \approx 9.24 \, \text{inches}
   \]

4. **Area of the Base**:
   The area of the base of the pyramid is the area of the square:
   \[
   \text{Area of the base} = a^2 = \frac{256}{3} \approx 85.33 \, \text{square inches}.
   \]

Thus, the area of the base is approximately **85.33 square inches**.

Would you like more details on any of the steps? 

Here are 5 questions that expand on this problem:
1. How is the slant height related to the side length of a regular square pyramid?
2. What are the general properties of a pyramid with equilateral triangles as lateral faces?
3. How would you calculate the surface area of this pyramid?
4. How does the altitude affect the shape and size of the base?
5. Could we solve this problem using trigonometric functions instead of the Pythagorean theorem?

### Tip: 
In pyramids, the slant height, base, and altitude often form right triangles, making the Pythagorean theorem a useful tool for calculations.

Find the area of the base of a regular square pyramid whose lateral faces are equilateral triangles and whose altitude is 8 in.

Learn how to find the area of the base of a regular square pyramid whose lateral faces are equilateral triangles and whose altitude is 8 inches. This problem involves key geometric concepts and the Pythagorean theorem. Suitable for students in Grades 9-11.

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