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