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To find the lateral area of this pyramid, we need to calculate the area of the three triangular faces that form the sides of the pyramid, excluding the base. Here’s the approach:

### Given Information
- The base of the pyramid is an equilateral triangle with each side measuring \(7\) inches.
- The slant height (the height from the base to the apex along one of the triangular faces) is \(4\) inches.

### Steps to Calculate the Lateral Area
1. **Calculate the Area of One Triangular Face**:
   Each triangular face has a base of \(7\) inches and a height (slant height) of \(4\) inches.
   
   The area of one triangular face is given by:
   \[
   \text{Area of one triangular face} = \frac{1}{2} \times \text{base} \times \text{slant height}
   \]
   Substituting the values:
   \[
   = \frac{1}{2} \times 7 \times 4 = \frac{1}{2} \times 28 = 14 \text{ square inches}
   \]

2. **Calculate the Total Lateral Area**:
   Since there are three identical triangular faces, the total lateral area is:
   \[
   \text{Total Lateral Area} = 3 \times 14 = 42 \text{ square inches}
   \]

### Answer
The lateral area of the pyramid is:
\[
\boxed{42 \text{ square inches}}
\]

Would you like further details or have any questions?

Here are 5 related questions to explore further:
1. How do you calculate the surface area of the entire pyramid, including the base?
2. What is the formula for finding the area of an equilateral triangle?
3. How would the calculation change if the slant height were different?
4. Can you find the volume of the pyramid if the height from the apex perpendicular to the base is given?
5. How would you determine the lateral area if the pyramid had a different polygonal base?

**Tip**: For pyramids with equilateral bases, knowing the slant height simplifies the calculation of lateral areas because each triangular face will have the same area.

Find the lateral area of the pyramid. The base is an equilateral triangle with each side measuring 7 inches, and the slant height is 4 inches.

Find the lateral area of a pyramid with an equilateral triangle base of 7 inches and a slant height of 4 inches. This geometry problem includes calculating the area of triangular faces and multiplying to find the total lateral area. Suitable for students in Grades 9-10.

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