Sure, let's solve this problem step-by-step.
1. Given Values:
- The area of the square base of the pyramid is [tex]\(64 \, \text{cm}^2\)[/tex].
- The height of the pyramid (perpendicular height from the base to the apex) is [tex]\(8 \, \text{cm}\)[/tex].
2. Find the Side Length of the Base:
Since the base is a square, the area of the base is given by:
[tex]\[
\text{Area} = \text{side length} \times \text{side length} = \text{side length}^2
\][/tex]
Therefore, to find the side length, we take the square root of the area:
[tex]\[
\text{side length} = \sqrt{64} = 8 \, \text{cm}
\][/tex]
3. Find the Slant Height:
The slant height of a pyramid is the distance from the apex to the midpoint of one of the sides of the base. This can be calculated using the Pythagorean theorem in the triangle formed by the height of the pyramid, half the side length of the base, and the slant height.
In this triangle:
- The perpendicular height of the pyramid is one leg: [tex]\(8 \, \text{cm}\)[/tex]
- Half the side length of the base is the other leg: [tex]\( \frac{8}{2} = 4 \, \text{cm}\)[/tex]
- The slant height is the hypotenuse, which we need to find.
According to the Pythagorean theorem:
[tex]\[
\text{slant height}^2 = (\text{half side length})^2 + (\text{height})^2
\][/tex]
Substituting the values:
[tex]\[
\text{slant height}^2 = 4^2 + 8^2 = 16 + 64 = 80
\][/tex]
To find the slant height, we take the square root:
[tex]\[
\text{slant height} = \sqrt{80} \approx 8.944 \, \text{cm}
\][/tex]
Thus, the side length of the base is [tex]\(8 \, \text{cm}\)[/tex] and the slant height of the pyramid is approximately [tex]\(8.944 \, \text{cm}\)[/tex].
