Area of Composite Figures

A composite figure (or irregular shape) is a shape made up of two or more simple geometric shapes, like rectangles, triangles, and parallelograms. To find the total area of these figures, you can either add simpler shapes together or subtract a missing piece from a larger shape.

Finding Area by Adding Shapes

When a figure is built by joining shapes together, follow these steps to find the total area:

1. Decompose the shape: Break the complex figure down into simpler shapes that you know how to find the area of.
2. Calculate each area: Use standard area formulas:
   • Rectangle: $A = l \times w$
   • Triangle: $A = \frac{1}{2}bh$
   • Parallelogram: $A = bh$
3. Add them up: Add the individual areas together to get the total area.

Example: Imagine a figure shaped like a house. It consists of a rectangle at the bottom and a triangle on top.

• The rectangle has a length of $8$ and a width of $5$. Its area is $8 \times 5 = 40$.
• The triangle has a base of $8$ and a height of $3$. Its area is $\frac{1}{2} \times 8 \times 3 = 12$.
• Total Area = $40 + 12 = 52$.

Finding Area by Subtracting (Shaded Regions)

Sometimes, a composite figure is described as a large shape with a smaller shape removed. This is common in "find the shaded region" problems.

1. Find the area of the larger outside shape.
2. Find the area of the unshaded (cut-out) shape.
3. Subtract the unshaded area from the larger area.

Example: You have a rectangular piece of paper that is $10$ by $6$. A small triangle with a base of $4$ and a height of $3$ is cut out of the middle. What is the area of the remaining paper?

• Area of the large rectangle: $10 \times 6 = 60$.
• Area of the cut-out triangle: $\frac{1}{2} \times 4 \times 3 = 6$.
• Remaining Area = $60 - 6 = 54$.

By carefully breaking apart irregular shapes, you can solve for the area of any composite figure!