To find the surface area of a square pyramid, we need to calculate the area of the base and the areas of the triangular faces.
Given:
- Four dimensions: 4 cm, 7 cm, 8 cm, and 10 cm.
We first need to clarify the dimensions:
- The base of the pyramid is typically square, so we’ll assume one of the given dimensions represents the side length of the square base.
- The height of the pyramid is used to find the area of the triangular faces.
Let’s assume that:
- The side length of the base (square) is 4 cm (this is a common assumption to start with a square).
- The height of the pyramid is 7 cm (we assume this is the vertical height from the base to the apex).
- The slant heights needed for the triangular faces will be determined based on the remaining lengths or calculated as required.
Step 1: Area of the Base (Square)
The area of the base (square): \[ \text{Area}_{\text{base}} = \text{side}^2 = 4 , \text{cm} \times 4 , \text{cm} = 16 , \text{cm}^2 \]
Step 2: Area of the Triangular Faces
Assuming we have a rectangular base and slant height; we will use the slant height as 8 cm for this calculation.
The area of one triangular face can be calculated using the formula: \[ \text{Area}_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height} \] Where:
- Base of the triangle = Side length of the square = 4 cm
- Height of the triangle = Slant height (assuming it to be 8 cm)
\[ \text{Area}_{\text{triangle}} = \frac{1}{2} \times 4 , \text{cm} \times 8 , \text{cm} = 16 , \text{cm}^2 \]
Since there are 4 triangular faces: \[ \text{Total Area}{\text{triangles}} = 4 \times \text{Area}{\text{triangle}} = 4 \times 16 , \text{cm}^2 = 64 , \text{cm}^2 \]
Step 3: Total Surface Area
Now, combine the area of the base with the total area of the triangular faces to find the total surface area: \[ \text{Total Surface Area} = \text{Area}{\text{base}} + \text{Total Area}{\text{triangles}} = 16 , \text{cm}^2 + 64 , \text{cm}^2 = 80 , \text{cm}^2 \]
Summary
- Area of the triangular faces: 64 cm²
- Area of the base (square): 16 cm²
- Total Surface Area: 80 cm²
Note: It's important to confirm the actual dimensions and definitions used in problems like this, as they have a big impact on the solution. If any dimensions represent something other than what’s assumed, please clarify for more specific calculations.