Certainly! Let's solve this step-by-step.
1. Identify the dimensions of the pyramid:
- The length of the base of the pyramid (which is a square) is 12 cm.
- The slant height of the pyramid is 15 cm.
2. Calculate the area of the base:
- The base of the pyramid is a square.
- The area of a square is given by: [tex]\(\text{side} \times \text{side}\)[/tex].
- In this case, each side of the base is 12 cm.
- Therefore, the area of the base is:
[tex]\[
12 \, \text{cm} \times 12 \, \text{cm} = 144 \, \text{cm}^2
\][/tex]
3. Calculate the area of the four triangular faces:
- Each triangular face has a base equal to the length of one side of the square base and a height equal to the slant height of the pyramid.
- The area of one triangle is given by: [tex]\(\frac{1}{2} \times \text{base} \times \text{height}\)[/tex].
- Base of each triangle = 12 cm (same as the length of the side of the square base).
- Height of each triangle = 15 cm (the slant height of the pyramid).
- Area of one triangular face:
[tex]\[
\frac{1}{2} \times 12 \, \text{cm} \times 15 \, \text{cm} = 90 \, \text{cm}^2
\][/tex]
- Since there are four such triangular faces, their total area is:
[tex]\[
4 \times 90 \, \text{cm}^2 = 360 \, \text{cm}^2
\][/tex]
4. Calculate the total surface area of the pyramid:
- The total surface area of the pyramid is the sum of the area of the base and the total area of the four triangular faces.
- Total surface area:
[tex]\[
144 \, \text{cm}^2 + 360 \, \text{cm}^2 = 504 \, \text{cm}^2
\][/tex]
Thus, the total surface area of the pyramid is [tex]\(504 \, \text{cm}^2\)[/tex].
