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To solve this problem, we need to find the slant height (lateral edge) \( S \) of a regular square pyramid with a given height and base diagonal.

### Given Data:
1. **Height of the pyramid** (\( SO \)) = 12 cm
2. **Diagonal of the square base** (\( AC \)) = 18 cm

### Solution:

1. **Finding the Half-Diagonal of the Base**:
   Since \( AC \) is the diagonal of the square base, we can calculate half of this diagonal (\( AO \)) to use as the base of the triangle \( \triangle SAO \).
   \[
   AO = \frac{AC}{2} = \frac{18 \, \text{cm}}{2} = 9 \, \text{cm}
   \]

2. **Using the Pythagorean Theorem**:
   The triangle \( \triangle SAO \) is a right triangle with:
   - \( SO \) (the height of the pyramid) as one leg,
   - \( AO \) (half the diagonal of the base) as the other leg,
   - \( SA \) (the slant height, which we are looking for) as the hypotenuse.

   According to the Pythagorean theorem:
   \[
   SA = \sqrt{SO^2 + AO^2}
   \]

3. **Substitute the Values**:
   \[
   SA = \sqrt{12^2 + 9^2}
   \]
   \[
   SA = \sqrt{144 + 81}
   \]
   \[
   SA = \sqrt{225}
   \]
   \[
   SA = 15 \, \text{cm}
   \]

### Answer:
The length of the lateral edge \( SA \) is **15 cm**.

Would you like more details on this solution or have any questions?

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Here are some related questions to deepen your understanding:

1. How would the lateral edge change if the height of the pyramid were different?
2. Can you calculate the length of the base side if the base is a square with diagonal 18 cm?
3. What would be the area of one of the triangular faces of this pyramid?
4. How do you find the surface area of this pyramid, including both the base and the lateral surfaces?
5. Can you calculate the volume of this pyramid using the given height and base area?

**Tip**: In geometry, always check if there's a right triangle involved to simplify calculations using the Pythagorean theorem.

Знайдіть бічне ребро правильної чотирикутної піраміди висота якої 12 см, а діагональ основи 18 см.

Learn how to find the lateral edge of a square pyramid with a height of 12 cm and a base diagonal of 18 cm. This problem involves using the Pythagorean theorem and square properties, suitable for students in Grades 9-10.

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