To find the area of this irregular figure, we can break it down into smaller, manageable shapes (rectangles and triangles). Let's proceed step by step:

Step 1: Break down the figure

The figure consists of:

1. A rectangle on the far right.
2. A trapezoid in the middle.
3. Another rectangle at the bottom left.
4. A triangle on the top left.

Step 2: Calculate the area of each shape

1. Right Rectangle:

• Dimensions: 4 inches (width) × 8 inches (height)
• Area: $4 \times 8 = 32 \, \text{square inches}$

2. Middle Trapezoid:

• The height is 5 inches (given).
• The two parallel sides are 1 inch (top) and 5 inches (bottom).
• Area of trapezoid formula: $\text{Area} = \frac{1}{2} \times (\text{Base}_1 + \text{Base}_2) \times \text{Height}$ Substituting the values: $\text{Area} = \frac{1}{2} \times (1 + 5) \times 5 = \frac{1}{2} \times 6 \times 5 = 15 \, \text{square inches}$

3. Bottom Left Rectangle:

• Dimensions: 5 inches (width) × 2 inches (height)
• Area: $5 \times 2 = 10 \, \text{square inches}$

4. Top Left Triangle:

• Base: 3 inches
• Height: 2 inches
• Area of triangle formula: $\text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height}$ Substituting the values: $\text{Area} = \frac{1}{2} \times 3 \times 2 = 3 \, \text{square inches}$

Step 3: Total Area

Now, add up the areas of all the parts: $\text{Total Area} = 32 + 15 + 10 + 3 = 60 \, \text{square inches}$

Final Answer:

The area of the figure is 60 square inches.

Would you like more details or clarifications? Here are some related questions to further your understanding:

1. How do you find the area of a trapezoid in general?
2. Can the area of an irregular shape always be broken into simpler parts?
3. What are the differences in calculating the area of polygons versus circles?
4. How would you calculate the area of a 3D object like a prism?
5. How does the concept of area extend to integration in calculus?

Tip: Always verify the breakdown of complex shapes by rechecking the dimensions and shape types before calculating the area.