Question

Follow the steps to find the Surface Area of the square pyramid.
Shape: Square Pyramid
8 cm and 14 cm
What is the area?
Square base: [?]
Triangles x4: [?]
Total SA: [?]

Answer 1

To find the surface area of a square pyramid, we'll need to calculate the area of the square base and the lateral area (the area of the four triangular faces). You've provided the dimensions: the base side length is 8 cm and the slant height (which is needed for the triangular area) is 14 cm.

### Step 1: Area of the Square Base
The area \(A\) of the square base can be found using the formula:
\[
A = s^2
\]
where \(s\) is the side length of the base.

For a base side length of 8 cm:
\[
A = 8 \, \text{cm} \times 8 \, \text{cm} = 64 \, \text{cm}^2
\]

### Step 2: Area of the Triangles
Each triangular face has a base equal to the side length of the square (\(s = 8 \, \text{cm}\)) and a height equal to the slant height (\(h = 14 \, \text{cm}\)). The area \(A_t\) of one triangular face can be calculated using the formula:
\[
A_t = \frac{1}{2} \times \text{base} \times \text{height}
\]
\[
A_t = \frac{1}{2} \times 8 \, \text{cm} \times 14 \, \text{cm} = \frac{1}{2} \times 112 \, \text{cm}^2 = 56 \, \text{cm}^2
\]

Since there are 4 triangular faces, the total area \(A_{total\_triangles}\) of the triangular sides is:
\[
A_{total\_triangles} = 4 \times A_t = 4 \times 56 \, \text{cm}^2 = 224 \, \text{cm}^2
\]

### Step 3: Total Surface Area
To find the total surface area \(SA\) of the pyramid, we add the area of the base and the area of the triangular faces:
\[
SA = A_{\text{base}} + A_{total\_triangles}
\]
\[
SA = 64 \, \text{cm}^2 + 224 \, \text{cm}^2 = 288 \, \text{cm}^2
\]

### Summary of Areas
- Square base area: \(64 \, \text{cm}^2\)
- Triangles area (4): \(224 \, \text{cm}^2\)
- Total Surface Area: \(288 \, \text{cm}^2\)

So the final answers are:
- Square base: **64 cm²**
- Triangles x4: **224 cm²**
- Total SA: **288 cm²**