Question

Find Surface area of a square pyramid with a of .5cm and I of .8cm

Answer 1

To find the surface area of a square pyramid, we need to consider the area of the square base as well as the areas of the four triangular faces.

You mentioned "a" and "I," which I will assume refer to:
- \( a \): length of a side of the square base (0.5 cm)
- \( I \): slant height of the pyramid (0.8 cm)

### Step 1: Area of the Base
The area \( A_{\text{base}} \) of the square base can be calculated using the formula:
\[
A_{\text{base}} = a^2
\]
Substituting \( a = 0.5 \, \text{cm} \):
\[
A_{\text{base}} = (0.5 \, \text{cm})^2 = 0.25 \, \text{cm}^2
\]

### Step 2: Area of the Triangular Faces
The area of one triangular face \( A_{\text{triangle}} \) can be calculated using the formula:
\[
A_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height}
\]
The base of each triangular face is the same as the side of the square base \( a = 0.5 \, \text{cm} \), and the height is the slant height \( I = 0.8 \, \text{cm} \).

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

Since there are four triangular faces, the total area of the triangular faces is:
\[
A_{\text{triangles}} = 4 \times A_{\text{triangle}} = 4 \times 0.2 \, \text{cm}^2 = 0.8 \, \text{cm}^2
\]

### Step 3: Total Surface Area
The total surface area \( A_{\text{total}} \) of the pyramid is the sum of the area of the base and the areas of the triangular faces:
\[
A_{\text{total}} = A_{\text{base}} + A_{\text{triangles}} = 0.25 \, \text{cm}^2 + 0.8 \, \text{cm}^2 = 1.05 \, \text{cm}^2
\]

Therefore, the surface area of the square pyramid is:
\[
\boxed{1.05 \, \text{cm}^2}
\]