Find the total surface area of a pyramid having a slant height of 8 cm and a base which is a square of side 4cm ( in cm )?

Options

A. 80
B. 64
C. 72
D. 84

Answer

To find the total surface area of a pyramid with a square base, we can follow these steps:

### Step 1: Identify the dimensions of the pyramid.
- The slant height (l) of the pyramid is given as 8 cm.
- The side length (a) of the square base is given as 4 cm.

### Step 2: Calculate the area of the base.
The area of the square base (A_base) can be calculated using the formula:
\[ A_{base} = a^2 \]
Substituting the value of a:
\[ A_{base} = 4^2 = 16 \, \text{cm}^2 \]

### Step 3: Calculate the area of one triangular face.
The area of one triangular face (A_triangle) can be calculated using the formula:
\[ A_{triangle} = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, the base of the triangle is the side of the square base (4 cm) and the height is the slant height (8 cm):
\[ A_{triangle} = \frac{1}{2} \times 4 \times 8 = \frac{32}{2} = 16 \, \text{cm}^2 \]

### Step 4: Calculate the total area of the triangular faces.
Since a square pyramid has 4 triangular faces, the total area of the triangular faces (A_triangles) is:
\[ A_{triangles} = 4 \times A_{triangle} = 4 \times 16 = 64 \, \text{cm}^2 \]

### Step 5: Calculate the total surface area of the pyramid.
The total surface area (TSA) of the pyramid is the sum of the area of the base and the total area of the triangular faces:
\[ TSA = A_{base} + A_{triangles} \]
Substituting the values:
\[ TSA = 16 + 64 = 80 \, \text{cm}^2 \]

### Final Answer:
The total surface area of the pyramid is **80 cm²**.

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