Complex and Composite Shapes

Break down complex shapes into simpler components for measurement

CAPS Grade 10 Mathematics

This topic focuses on breaking down complex shapes into simpler components for measurement. Learn to calculate perimeter, area, and volume of composite shapes.

Basic Shape Formulas Review

Square

P = 4a | A = a²

Rectangle

P = 2(l+w) | A = l × w

Triangle

P = a+b+c | A = ½×b×h

Circle

C = 2πr | A = πr²

Parallelogram

P = 2(a+b) | A = b × h

Trapezium

A = ½×(a+b)×h

Quiz 1 - Basic Formulas

What is the area formula for a triangle

A) l × w

B) ½ × b × h

C) πr²

D) s²

Learning Outcomes

Identify and describe complex and composite shapes
Decompose complex shapes into simpler geometric figures
Calculate perimeter and area of composite shapes
Calculate surface area and volume of composite 3D objects
Solve real-world problems involving complex shapes

L-Shape Composite Figure

L-Shape decomposed into two rectangles: A (8×4) and B (3×2)

3. Area Calculations: Additive Method

L-Shaped Figure

Problem

L-shape: total 8m × 6m with 4m × 3m removed from corner. Find area.

Solution

Rectangle A: 8m × 4m = 32 m²

Rectangle B: 3m × 2m = 6 m²

Total Area = 32 + 6 = 38 m²

Shape with Triangle

Problem

Rectangle 10cm × 6cm with triangle on top (base 10cm, height 4cm). Find area.

Solution

Rectangle: 10 × 6 = 60 cm²

Triangle: ½ × 10 × 4 = 20 cm²

Total = 80 cm²

Quiz 2 - Additive Method

Rectangle 8m × 5m plus triangle base 8m, height 3m on top. Total area

A) 40 m²

B) 52 m²

C) 64 m²

D) 72 m²

Rectangle with Semicircle

Composite shape: Rectangle with a semicircle on top

4. Area Calculations: Subtractive Method

Shape with Hole

Problem

Rectangle 12m × 8m with semicircle (radius 2m) removed. Find area.

Solution

Rectangle: 12 × 8 = 96 m²

Semicircle: ½ × π × 2² = 2π ≈ 6.28 m²

Total = 96 - 6.28 = 89.72 m²

Complex Cut-out

Problem

Square 10cm × 10cm with quarter circle (radius 5cm) removed. Find area.

Solution

Square: 10² = 100 cm²

Quarter circle: ¼ × π × 5² = 6.25π ≈ 19.63 cm²

Total = 100 - 19.63 = 80.37 cm²

Quiz 3 - Composite 3D

Cube side 8cm with cylindrical hole r=1cm through center. Remaining volume (π≈3.14)

A) 487 cm³

B) 512 cm³

C) 500 cm³

D) 450 cm³

5. Perimeter of Complex Shapes

Perimeter Rules

Add all outer sides
Don't include internal division lines
For curved parts, use circumference formulas

L-shape Example

P = 8 + 6 + 3 + 2 + 4 + 5 = 28 m

Semicircle Example

P = 12 + 8 + 12 + π×2 = 32 + 6.28 = 38.28 m

Practice & Assess

Test your knowledge with these interactive games.

Match - Calculation Method

Additive

Add areas of parts

Subtractive

Subtract holes

Perimeter

Sum of outer edges

Composite 3D

Combine 3D shapes

Fill - Trapezium Area

A = ½ × (a + b) × ___

7. Real-World Applications

Painting a Wall

Wall 4m × 3m, window 1m × 1.5m, door 0.8m × 2m. Paintable area

12 - 1.5 - 1.6 = 8.9 m²

Garden Fencing

L-shaped garden 8m×6m and 4m×3m. Fencing needed

Perimeter = 8+6+3+2+4+5 = 28 m

Container Volume

Cylindrical tank (r=2m, h=3m) with conical bottom (h=1m). Total volume

12π + 1.33π = 13.33π m³

Practice Questions

T-shape: vertical 6×4, horizontal 8×2. Find area.

Circle r=5cm with square side=3cm cut out. Remaining area

Swimming pool: rectangle 10×4 with semicircle r=2 at each end. Perimeter

Summary of Key Concepts

Additive Method: Break shape into parts and add

Subtractive Method: Start with outer shape, subtract holes

Perimeter: Add only outer edges

Unit Consistency: Convert all to same unit first

Key Terms

Composite Decompose Additive Subtractive Perimeter Area Volume Surface Area Semicircle Quarter Circle Hemisphere

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