Let's go through each part of the problem step by step:

Given Information:

• The base of the pyramid is a square.
• The length of each side of the square is 7 inches.
• The height of each triangular face (slant height) is 5.4 inches.
• The net includes 4 congruent triangular lateral faces and the square base.

Part (a): Find the area of the base of the pyramid.

The base of the pyramid is a square with side length 7 inches. The area of a square is calculated as:

$\text{Area of the base} = \text{side}^2 = 7^2 = 49 \, \text{square inches}$

Part (b): Find the area of one lateral face of the pyramid.

Each lateral face is a triangle with a base of 7 inches (same as the side of the square) and a height (slant height) of 5.4 inches. The area of a triangle is:

$\text{Area of one triangle} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 7 \times 5.4 = 18.9 \, \text{square inches}$

Part (c): Find the lateral surface area of the pyramid (without the base).

The lateral surface area is the combined area of all four triangular faces. Since all four triangles are congruent, we can multiply the area of one triangle by 4:

$\text{Lateral surface area} = 4 \times 18.9 = 75.6 \, \text{square inches}$

Part (d): Find the total surface area of the pyramid.

The total surface area includes the lateral surface area (from part c) plus the area of the base (from part a):

$\text{Total surface area} = \text{Lateral surface area} + \text{Area of the base} = 75.6 + 49 = 124.6 \, \text{square inches}$

Summary:

• (a) Area of the base = 49 in²
• (b) Area of one lateral face = 18.9 in²
• (c) Lateral surface area = 75.6 in²
• (d) Total surface area = 124.6 in²

Do you need further details or explanations on any part of this? Here are 5 related questions to expand on the concept:

1. What would happen to the lateral surface area if the slant height were increased?
2. How would you find the volume of this pyramid?
3. Can you generalize the formula for the surface area of any square pyramid?
4. If the side of the base were doubled, how would the total surface area change?
5. How would you calculate the surface area if the pyramid had a rectangular base?

Tip:

For any polygonal pyramid, the lateral surface area can always be found by calculating the area of one face and multiplying by the number of sides in the base.