Area of Composite Figures

Free sample questions, a clear explanation, and 5 practice skills with an AI tutor that guides without giving the answer away.

Concept Review

Area of Composite Figures: Building Block Geometry

Imagine you're an architect designing a house with an L-shaped living room. How would you calculate the carpet needed to cover that oddly-shaped floor? Welcome to the world of composite figures — shapes made by combining simpler shapes like rectangles and triangles.

The secret to finding the area of any composite figure is surprisingly simple: break it down into familiar pieces. Every complex shape is just a puzzle of rectangles and triangles waiting to be solved.

The Decomposition Strategy

Let's tackle a real example. Picture a house-shaped figure: a rectangle (15 feet wide, 12 feet tall) topped with a triangle (same 15-foot base, 8 feet tall).

Step-by-Step Calculation:

Rectangle area: 15 × 12 = 180 square feet

Triangle area: ½ × 15 × 8 = 60 square feet

Total area: 180 + 60 = 240 square feet

This same strategy works whether you're dealing with swimming pools with diving platforms, school playgrounds with multiple sections, or any shape that combines our basic building blocks.

💡 The Subtraction Surprise

Here's something that might blow your mind: sometimes it's easier to subtract areas instead of adding them!

Imagine finding the area of a rectangular garden with a triangular flower bed cut out of the middle. Calculate the full rectangle, then subtract the triangle. Same answer, less work!

Multiple Approaches, Same Answer

The beauty of composite figures is that there's often more than one way to break them apart. An L-shaped figure could be split into two rectangles vertically or horizontally. Both methods will give you the exact same area — it's like solving the same puzzle with different strategies.

The key is identifying clean lines where one shape ends and another begins. Look for right angles, parallel lines, and familiar shapes hiding within the complex figure.

🔑 Key Takeaway

Just like that L-shaped living room, every composite figure is simply a combination of shapes you already know how to handle. Master rectangles and triangles, and you can carpet any room — no matter how uniquely designed.

Sample questions

1. A figure is made of a rectangle 8 cm by 5 cm with a triangle on top having base 8 cm and height 3 cm. What is the total area?

✓ 52 cm²? Rectangle area = 40 cm², triangle area = ½ × 8 × 3 = 12 cm², total = 52 cm²

○ 52 cm²

○ 40 cm²

○ 52 cm²

Answer: 52 cm²? Rectangle area = 40 cm², triangle area = ½ × 8 × 3 = 12 cm², total = 52 cm² — Add the areas: rectangle (40) + triangle (12) = 52 cm².

2. Find the area of an L-shaped figure that can be split into two rectangles: one 6 m by 4 m and another 3 m by 2 m.

○ 30 m²

✓ 30 m²? 6×4=24, 3×2=6, total=30

○ 24 m²

○ 30 m²

Answer: 30 m²? 6×4=24, 3×2=6, total=30 — Add the areas of the two rectangles: 24 + 6 = 30 m².

3. A figure consists of a rectangle 10 ft by 6 ft with a right triangle cut out from one corner. The triangle has legs 4 ft and 3 ft. What is the area of the remaining figure?

○ 54 ft²

○ 60 ft²

✓ 54 ft²? Rectangle area = 60 ft², triangle area = ½ × 4 × 3 = 6 ft², remaining = 60 - 6 = 54 ft²

○ 54 ft²

Answer: 54 ft²? Rectangle area = 60 ft², triangle area = ½ × 4 × 3 = 6 ft², remaining = 60 - 6 = 54 ft² — Subtract the triangle area from the rectangle area.

Skills in this topic

Calculate the area of figures composed of rectangles and triangles
Calculate the area of figures composed of polygons and semicircles
Calculate the area of a shaded region by subtracting the inner shape area
Decompose complex polygons into familiar shapes to find the area
Solve real-world problems involving the area of complex floor plans

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