Area of Triangles & Parallelograms • Topic 3 of 3

Area of Irregular Polygons by Composing/Decomposing

What is an Irregular Polygon?

An irregular polygon is a shape that is not a standard shape like a triangle, square, or rectangle. The sides are not all equal, and angles may vary.

Two Methods to Find Area:

MethodDescriptionWhen to Use
DecomposingBreak shape into smaller shapes (triangles, rectangles)Shape has "indents" or can be split easily
ComposingAdd extra shapes to form a larger shape, then subtractShape has "missing pieces"

Strategy:

  1. Look for ways to split the shape into rectangles, triangles, and parallelograms
  2. Find area of each part
  3. Add all areas together
DECOMPOSING IRREGULAR POLYGON:

 L-shaped figure:
 
 ┌─────────┐
 │ │
 │ A │ ┌───┐
 │ │ │ C │
 │ │ │ │
 ├─────┐ │ │ │
 │ B │ │ │ │
 │ │ │ │ │
 └─────┴───┘ └───┘
 
 Method 1: Split into rectangles A and B
 Area = A + B
 
 Method 2: Split into rectangles A, B, and C
 Area = A + B + C


DECOMPOSING INTO TRIANGLES AND RECTANGLES:

 Irregular pentagon:
 
 /│
 / │ 
 / │ 
 / │ 
 / │ 
 └─────┴─────┘
 
 Split into:
 - Rectangle in middle
 - Triangle on top
 - Triangles on sides (or just one triangle if symmetrical)


COMPOSING (ADD THEN SUBTRACT):

 Shape with a "hole":
 
 ┌─────────────┐
 │ │
 │ ┌─────┐ │
 │ │ │ │
 │ └─────┘ │
 │ │
 └─────────────┘
 
 Area = Big rectangle - Small rectangle


COMPLEX SHAPE EXAMPLE:

 Arrow shape:
 
 ┌───┐
 │ │
 ┌───┼───┼───┐
 │ │ │ │
 └───┼───┼───┘
 │ │
 └───┘
 
 Decompose into:
 - Center rectangle
 - Left rectangle
 - Right rectangle
 - Bottom rectangle
Solved Examples
1
Worked Example

Find the area of this L-shaped figure (all measurements in cm):

  • Top rectangle: 8 cm × 3 cm
  • Bottom rectangle: 5 cm × 4 cm (overlaps with top by 3 cm)
Solution
  • Top rectangle area = 8 × 3 = 24 cm²
  • Bottom rectangle width = 5 cm, height = 4 cm, area = 20 cm²
  • But the overlapping corner (3×3=9) is counted twice if we just add
  • Better: Split into non-overlapping rectangles
  • Rectangle A: 8 × 3 = 24
  • Rectangle B: 5 × 4 = 20
  • No overlap if split differently!
  • Total = 24 + 20 = 44
  • Answer: 44 cm²
2
Worked Example

Find the area of a shape made of a 6×4 rectangle with a 3×2 square attached on top.

Solution
  • Rectangle: 6 × 4 = 24 cm²
  • Square: 3 × 2 = 6 cm²
  • Total = 24 + 6 = 30
  • Answer: 30 cm²
3
Worked Example

A shape consists of a triangle (base 8 cm, height 5 cm) attached to a rectangle (8 cm × 4 cm). Find total area.

Solution
  • Triangle area = ½ × 8 × 5 = 20 cm²
  • Rectangle area = 8 × 4 = 32 cm²
  • Total = 20 + 32 = 52
  • Answer: 52 cm²

Key Points

  • Decompose = break into smaller shapes
  • Compose = add pieces to make larger shape, then subtract
  • Label all dimensions clearly
  • Add areas of all parts
  • Check for overlaps (don't count twice!)
Quick Check — 4 MCQs
Tap an option to check your answer0 / 4
Q1.To find the area of an irregular polygon, split it into:
Explanation: Known shapes.
Q2.The total area is the ___ of the parts.
Explanation: Sum.
Q3.To find a remaining area, you ___ the removed region.
Explanation: Subtract it.
Q4.When adding areas, always use the same:
Explanation: Same unit.
Practice more MCQs → 📝 Assignment · 35 marks