To solve this problem, we need to find the total surface area of a right regular pyramid with a square base.

Step 1: Formula for Surface Area of a Pyramid

The total surface area $A$ of a regular pyramid with a square base is given by the formula:

$A = B + L$

Where:

• $B$ is the area of the base.
• $L$ is the lateral surface area (sum of the areas of the triangular faces).

Step 2: Base Area

Since the base is a square, the area $B$ is calculated as:

$B = s^2$ Where $s$ is the side length of the square base. From the diagram, the side length $s = 6 \, \text{ft}$.

So, the base area $B$ is:

$B = 6^2 = 36 \, \text{ft}^2$

Step 3: Lateral Surface Area

The lateral surface area consists of four triangular faces. The area of one triangular face is:

$A_{\text{triangle}} = \frac{1}{2} \times \text{base of the triangle} \times \text{slant height}$

From the diagram, the base of each triangular face is $6 \, \text{ft}$ (the side length of the square), and the slant height $l = 5 \, \text{ft}$.

Thus, the area of one triangular face is:

$A_{\text{triangle}} = \frac{1}{2} \times 6 \times 5 = 15 \, \text{ft}^2$

Since there are 4 triangular faces, the total lateral surface area $L$ is:

$L = 4 \times 15 = 60 \, \text{ft}^2$

Step 4: Total Surface Area

Now, we can find the total surface area by adding the base area and the lateral surface area:

$A = B + L = 36 + 60 = 96 \, \text{ft}^2$

Answer:

The total surface area of the pyramid is $96 \, \text{ft}^2$, so the correct answer is:

A. $96 \, \text{ft}^2$

Would you like further clarification or details on any step? Here are some related questions:

1. How do you calculate the surface area for pyramids with different shaped bases?
2. What is the difference between slant height and vertical height in pyramids?
3. How do you find the volume of a regular pyramid?
4. Can this method be applied to pyramids with non-square bases?
5. How does the formula for lateral area change if the base is not regular?

Tip: When working with pyramids, make sure you know whether you’re using the vertical height or slant height, as they are often confused in calculations.