Complex Figures / Composite Shapes

TipLearning Objectives
  • Break composite shapes into simpler parts.
  • Compute area and perimeter of combined figures.
  • Identify hidden right triangles and rectangles.

Key Ideas

A composite shape is made from multiple basic shapes: triangles, rectangles, semicircles.

Strategy:
1. Break apart using straight cuts.
2. Compute each area or perimeter.
3. Add or subtract depending on shape.

Composite figure made of a rectangle, semicircle, and triangle—used to illustrate breaking a complex shape into simpler parts.

Common Problem Types

Split Into Triangles and Rectangles

Identify right angles.

Area With Cut-Outs

Subtract area of missing region.

Finding Missing Measurements

Use Pythagorean Theorem or similarity.

Perimeter of Composite Shapes

Only boundary counts.

Real-World Shapes

Floorplans, gardens, packaging.

Strategies

  • Draw lines to create rectangles/triangles.
  • Label all known lengths.
  • Use Pythagorean Theorem for diagonal cuts.
  • Watch which sides are interior vs. boundary.

Worked Examples

Example 1 — Composite Area

A shape consists of a 10×6 rectangle plus a right triangle with legs 6 and 8.

Rectangle area = 60
Triangle area = 24
Total = 84

Example 2 — Missing side

Diagonal across 6×8 rectangle:
\[ c=\sqrt{6^2+8^2}=10. \]

WarningCommon Mistakes
  • Counting inside edges in perimeter.
  • Forgetting to subtract cut-out areas.
  • Ignoring right triangles created by diagonals.
  • Using wrong base/height in triangles.

Practice Problems

  1. Rectangle 12×5 + triangle with base 12 and height 4.
  2. Find diagonal of 9×12 rectangle.
  3. Composite: big rectangle area 50, small cut-out area 12 → total?
  4. A composite has perimeter of outer edges only — identify which edges count.
  1. \(60 + 24 = 84\)
  2. \(d = 15\)
  3. \(50 - 12 = 38\)
  4. Include only boundary edges.

Summary

  • Decompose shapes into basic parts.
  • Add areas or subtract for cut-outs.
  • Use triangles and rectangles to simplify.
  • Boxing and slicing simplifies complex shapes.
  • Pythagorean Theorem appears frequently.