Let's work through the problem to understand Lin's reasoning and calculate the exact area of the octagon.

Step 1: Understanding the Octagon's Shape

The octagon described consists of:

• 4 straight sides, each measuring 4 inches in length.
• 4 angled sides, each with a height of 3 inches and a base of 3 inches.

The octagon appears to be symmetric, and it has 8 sides total: 4 straight horizontal/vertical sides and 4 angled diagonal sides.

Step 2: Simplifying the Problem

The simplest way to find the area of the octagon is to split it into recognizable shapes such as rectangles and triangles.

Rectangles:

The middle section of the octagon is a rectangle formed by the 4 straight sides. The length of the rectangle is $4 + 3 + 3 = 10$ inches (since the diagonal sides on either end add 3 inches each), and the width is 4 inches.

• The area of the rectangle is: $A_{\text{rectangle}} = \text{length} \times \text{width} = 10 \times 4 = 40 \text{ square inches}$

Triangles:

The angled sides can be represented by 4 right triangles, each with:

• A base of 3 inches,

• A height of 3 inches.

• The area of one triangle is: $A_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 3 \times 3 = 4.5 \text{ square inches}$ Since there are 4 triangles, the total area for the 4 triangles is: $A_{\text{triangles}} = 4 \times 4.5 = 18 \text{ square inches}$

Step 3: Total Area of the Octagon

The total area of the octagon is the sum of the area of the rectangle and the area of the 4 triangles: $A_{\text{octagon}} = A_{\text{rectangle}} + A_{\text{triangles}} = 40 + 18 = 58 \text{ square inches}$

Step 4: Evaluating Lin’s Estimate

Lin reasoned that the area of the octagon must be less than 100 square inches. Since we calculated the exact area to be 58 square inches, Lin's reasoning is correct. The area is indeed less than 100 square inches.

Final Answer:

The exact area of the octagon is 58 square inches.

Would you like further details or have any questions? Here are some related questions for you:

1. How do you derive the area of a regular octagon using side lengths?
2. What are the properties of an irregular octagon?
3. How do you calculate the area of an octagon inscribed in a circle?
4. How does the formula for the area of a triangle change with different base and height?
5. What are other ways to estimate the area of an octagon?

Tip: Decomposing complex shapes into basic geometric figures like triangles and rectangles can simplify area calculations greatly.