To find the lateral area of a regular square pyramid with base edges of length 24 units and a perpendicular height of 5 units, we follow these steps:
1. Determine the slant height:
- The slant height ([tex]\(l\)[/tex]) in a pyramid is the distance from the midpoint of one of the base edges to the apex of the pyramid along the triangular face.
- Since the base edge is 24 units, half of this edge length is [tex]\( \frac{24}{2} = 12 \)[/tex] units.
- Using the Pythagorean theorem to find the slant height [tex]\( l \)[/tex]:
[tex]\[
l = \sqrt{ ( \text{half of base edge} )^2 + (\text{height of pyramid})^2 }
\][/tex]
[tex]\[
l = \sqrt{ 12^2 + 5^2 }
\][/tex]
[tex]\[
l = \sqrt{ 144 + 25 }
\][/tex]
[tex]\[
l = \sqrt{ 169 }
\][/tex]
[tex]\[
l = 13 \text{ units}
\][/tex]
2. Calculate the lateral area of one triangular face:
- Each triangular face of the pyramid has a base of 24 units and a slant height of 13 units.
- The area of one triangular face ([tex]\(A\)[/tex]) is given by:
[tex]\[
A = \frac{1}{2} \times \text{base} \times \text{slant height}
\][/tex]
[tex]\[
A = \frac{1}{2} \times 24 \times 13
\][/tex]
[tex]\[
A = \frac{1}{2} \times 312
\][/tex]
[tex]\[
A = 156 \text{ square units}
\][/tex]
3. Calculate total lateral area of the pyramid:
- A square pyramid has 4 triangular faces.
- So, the total lateral area ([tex]\(LA\)[/tex]) is:
[tex]\[
LA = 4 \times \text{lateral area of one triangular face}
\][/tex]
[tex]\[
LA = 4 \times 156
\][/tex]
[tex]\[
LA = 624 \text{ square units}
\][/tex]
Hence, the lateral area of the square pyramid is:
C. 624 units [tex]$^2$[/tex]
