Here is an octagon.

An octagon with straight sides that are 4 inches long, and angled sides that are both 3 inches high and 3 inches wide.
While estimating the area of the octagon, Lin reasoned that it must be less than 100 square inches. Do you agree? Explain your reasoning.
Find the exact area of the octagon. Show your reasoning.

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.

An octagon with straight sides that are 4 inches long, and angled sides that are both 3 inches high and 3 inches wide. While estimating the area of the octagon, Lin reasoned that it must be less than 100 square inches. Do you agree? Explain your reasoning. Find the exact area of the octagon.

Learn how to calculate the exact area of an irregular octagon with straight sides of 4 inches and angled sides of 3 inches by decomposing it into simple geometric shapes like rectangles and triangles. This problem covers fundamental geometry concepts and is suitable for Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation