To determine the lateral surface area of the pyramid from the image provided:

Given Information:

• The base of the pyramid is an equilateral triangle.
• Each triangular face (lateral face) has:
  • A base length of 4 inches.
  • A slant height of 8 inches (the height of each triangular face).

The lateral surface area includes the areas of the three triangular faces around the base.

Step 1: Area of One Triangular Face

The area of a triangle is given by:

$\text{Area of one triangle} = \frac{1}{2} \times \text{base} \times \text{height}$

Substituting the values: $\text{Area of one triangle} = \frac{1}{2} \times 4 \times 8 = 16 \, \text{square inches}.$

Step 2: Total Lateral Surface Area

Since there are 3 identical triangular faces: $\text{Total lateral surface area} = 3 \times \text{Area of one triangle}.$

Substituting the value: $\text{Total lateral surface area} = 3 \times 16 = 48 \, \text{square inches}.$

Final Answer:

The lateral surface area of the pyramid is 48 square inches.

Let me know if you’d like further details or explanations! 😊

Related Questions:

1. How is the total surface area of the pyramid calculated (including the base)?
2. What is the perimeter of the base of the pyramid?
3. How do you find the height of the equilateral triangular base using its side length?
4. What would the lateral surface area be if the slant height were doubled?
5. Can you use trigonometry to verify the slant height of the triangular face?

Tip:

Always double-check the base type (equilateral, isosceles, etc.) in pyramid problems to avoid calculation errors!