Question

Apply the Pythagorean Theorem to determine the height of the square pyramid, h . Round the answer to the nearest tenth. Responses

Answer 1

To determine the height of a square pyramid using the Pythagorean theorem, we typically need two key pieces of information: the length of the slant height (the distance from the apex of the pyramid to the midpoint of a side of the base) and the length of a side of the base.

Here's a general step-by-step guide on how to find the height \( h \) of a square pyramid:

1. **Identify the Length of the Base:** Let \( a \) be the length of a side of the base of the pyramid.

2. **Identify the Slant Height:** Let \( l \) be the slant height of the pyramid.

3. **Determine Half the Base Length:** Since the base is a square, half of the side length is \( \frac{a}{2} \).

4. **Set Up the Right Triangle:** The height \( h \), half the base length \( \frac{a}{2} \), and the slant height \( l \) form a right triangle. According to the Pythagorean theorem:
\[
l^2 = h^2 + \left(\frac{a}{2}\right)^2
\]

5. **Solve for Height \( h \):** Rearranging the equation for height gives:
\[
h^2 = l^2 - \left(\frac{a}{2}\right)^2
\]
\[
h = \sqrt{l^2 - \left(\frac{a}{2}\right)^2}
\]

6. **Calculate \( h \):** Plug in the values of \( l \) and \( a \) to compute \( h \), and then round to the nearest tenth.

### Example Calculation

For example, let’s say the side length of the base \( a = 6 \) units, and the slant height \( l = 7 \) units.

1. Half of the base length:
\[
\frac{a}{2} = \frac{6}{2} = 3 \text{ units}
\]

2. Apply the Pythagorean theorem:
\[
h^2 = 7^2 - 3^2 = 49 - 9 = 40
\]
\[
h = \sqrt{40} \approx 6.324
\]

3. Round \( h \) to the nearest tenth:
\[
h \approx 6.3 \text{ units}
\]

If you have specific values for the base length and slant height, please provide them, and I can help you calculate \( h \) precisely.