Question

Find surface area of square pyramid with a side measuring .5cm and a slant height of .8cm

Answer 1

To find the surface area of a square pyramid, you need to calculate both the base area and the lateral surface area.

1. **Base Area**:
The base of the pyramid is a square. The area \(A\) of a square is given by:
\[
A = \text{side}^2
\]
For a side length of \(0.5 \, \text{cm}\):
\[
A = (0.5 \, \text{cm})^2 = 0.25 \, \text{cm}^2
\]

2. **Lateral Surface Area**:
The lateral surface area consists of the areas of the four triangular faces. The area \(A_t\) of one triangle can be calculated using the formula:
\[
A_t = \frac{1}{2} \times \text{base} \times \text{height}
\]
Here, the base of each triangular face is the side of the square pyramid, which is \(0.5 \, \text{cm}\), and the height of the triangle is the slant height of the pyramid.

Therefore, the area of one triangular face is:
\[
A_t = \frac{1}{2} \times 0.5 \, \text{cm} \times 0.8 \, \text{cm} = \frac{1}{2} \times 0.4 \, \text{cm}^2 = 0.2 \, \text{cm}^2
\]

Since there are four triangular faces, the total lateral surface area \(A_l\) is:
\[
A_l = 4 \times A_t = 4 \times 0.2 \, \text{cm}^2 = 0.8 \, \text{cm}^2
\]

3. **Total Surface Area**:
The total surface area \(A_{total}\) of the pyramid is the sum of the base area and the lateral surface area:
\[
A_{total} = \text{Base Area} + \text{Lateral Surface Area}
\]
\[
A_{total} = 0.25 \, \text{cm}^2 + 0.8 \, \text{cm}^2 = 1.05 \, \text{cm}^2
\]

Thus, the surface area of the square pyramid is \( \boxed{1.05 \, \text{cm}^2} \).