Area of Composite Figures

A composite figure is a shape made up of two or more basic shapes. To find the area, you break it into simpler shapes, find each area, then add them together!

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What Is a Composite Figure?

A composite figure (also called a complex figure) is created by combining basic shapes like:

• Rectangles
• Triangles
• Circles
• Trapezoids
• Semicircles

Real-world examples:

• Floor plans of houses
• Gardens with multiple sections
• Swimming pools with different depths
• Letters and logos

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Review: Area Formulas for Basic Shapes

Rectangle: A = length × width = lw

Square: A = side × side = s²

Triangle: A = (1/2) × base × height = (1/2)bh

Parallelogram: A = base × height = bh

Trapezoid: A = (1/2) × (base₁ + base₂) × height = (1/2)(b₁ + b₂)h

Circle: A = πr² (where r = radius)

Semicircle: A = (1/2)πr² (half of a circle)

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Strategy for Finding Area of Composite Figures

Step 1: Break the figure into basic shapes

• Look for rectangles, triangles, circles, etc.
• Draw lines to separate shapes

Step 2: Find the area of each basic shape

• Use appropriate formula for each piece
• Label dimensions carefully

Step 3: Add all the areas together

• Total Area = Area₁ + Area₂ + Area₃ + ...

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Method 1: Addition Method

Break the figure into parts and ADD the areas.

Example 1: L-Shaped Figure

An L-shaped figure can be split into two rectangles:

• Top rectangle: 8 ft × 3 ft
• Bottom rectangle: 5 ft × 4 ft

Solution:

Area of top rectangle: A₁ = 8 × 3 = 24 ft²

Area of bottom rectangle: A₂ = 5 × 4 = 20 ft²

Total Area: A = 24 + 20 = 44 ft²

Answer: 44 ft²

Example 2: Rectangle with Triangle on Top

A house-shaped figure: rectangle base with triangular roof

• Rectangle: 12 m × 8 m
• Triangle: base = 12 m, height = 5 m

Solution:

Rectangle area: A₁ = 12 × 8 = 96 m²

Triangle area: A₂ = (1/2) × 12 × 5 = 30 m²

Total Area: A = 96 + 30 = 126 m²

Answer: 126 m²

Example 3: Rectangle with Semicircle

A shape with rectangle (10 cm × 6 cm) and semicircle on one end (diameter = 6 cm, radius = 3 cm)

Solution:

Rectangle area: A₁ = 10 × 6 = 60 cm²

Semicircle area: A₂ = (1/2)π(3²) = (1/2)π(9) = 4.5π ≈ 14.14 cm²

Total Area: A = 60 + 14.14 ≈ 74.14 cm²

Answer: About 74.14 cm²

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Method 2: Subtraction Method

Sometimes it's easier to find a large shape's area, then SUBTRACT a missing piece!

Example 1: Rectangle with Corner Cut Out

A 10 in × 8 in rectangle with a 3 in × 2 in rectangle cut from one corner.

Solution:

Large rectangle area: A₁ = 10 × 8 = 80 in²

Cut-out rectangle area: A₂ = 3 × 2 = 6 in²

Remaining Area: A = 80 - 6 = 74 in²

Answer: 74 in²

Example 2: Circle with Hole

A circular disc with outer radius 10 cm and a circular hole with radius 4 cm.

Solution:

Outer circle area: A₁ = π(10²) = 100π cm²

Inner circle (hole) area: A₂ = π(4²) = 16π cm²

Remaining Area: A = 100π - 16π = 84π ≈ 263.89 cm²

Answer: About 263.89 cm²

Example 3: Rectangle with Triangle Removed

A 15 ft × 10 ft rectangle with a triangle (base = 6 ft, height = 4 ft) removed from one corner.

Solution:

Rectangle area: A₁ = 15 × 10 = 150 ft²

Triangle area: A₂ = (1/2) × 6 × 4 = 12 ft²

Remaining Area: A = 150 - 12 = 138 ft²

Answer: 138 ft²

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Finding Missing Dimensions

Sometimes you need to calculate missing dimensions before finding areas!

Example: T-Shaped Figure

A T-shaped figure:

• Top horizontal bar: 12 cm × 3 cm
• Vertical stem: ? cm × 4 cm
• Total height: 10 cm

Step 1: Find missing dimension Vertical stem height = 10 - 3 = 7 cm

Step 2: Find areas Top bar: 12 × 3 = 36 cm² Vertical stem: 7 × 4 = 28 cm²

Step 3: Add Total = 36 + 28 = 64 cm²

Answer: 64 cm²

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Working with Multiple Shapes

Example: Complex Floor Plan

A room has:

• Main rectangular area: 20 ft × 15 ft
• Alcove (small rectangle): 8 ft × 5 ft
• Circular bay window: radius = 4 ft (semicircle)

Solution:

Main room: A₁ = 20 × 15 = 300 ft²

Alcove: A₂ = 8 × 5 = 40 ft²

Bay window (semicircle): A₃ = (1/2)π(4²) = 8π ≈ 25.13 ft²

Total Area: A = 300 + 40 + 25.13 = 365.13 ft²

Answer: About 365 ft²

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Real-World Applications

Landscaping

Problem: A garden has a rectangular section (15 m × 10 m) and a triangular section (base = 10 m, height = 6 m). How much area needs to be planted?

Solution: Rectangle: 15 × 10 = 150 m² Triangle: (1/2) × 10 × 6 = 30 m² Total: 150 + 30 = 180 m²

Answer: 180 m²

Painting

Problem: A wall shaped like a house (rectangle with triangle on top) needs paint. Rectangle is 12 ft × 8 ft, triangle has base 12 ft and height 3 ft. One gallon covers 50 ft². How many gallons needed?

Solution: Rectangle: 12 × 8 = 96 ft² Triangle: (1/2) × 12 × 3 = 18 ft² Total: 96 + 18 = 114 ft² Gallons: 114 ÷ 50 = 2.28 → Need 3 gallons

Answer: 3 gallons

Flooring

Problem: An L-shaped room needs tile. One section is 14 ft × 10 ft, the other is 8 ft × 6 ft. Tiles cost $3 per square foot. What's the total cost?

Solution: Section 1: 14 × 10 = 140 ft² Section 2: 8 × 6 = 48 ft² Total area: 188 ft² Cost: 188 × $3 =$564

Answer: $564

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Using Both Addition and Subtraction

Example: Swimming Pool

A rectangular pool (20 m × 10 m) has a semicircular extension on one end (radius = 5 m), but a rectangular shallow area (4 m × 3 m) is excluded from the total.

Solution:

Main pool: A₁ = 20 × 10 = 200 m²

Semicircular extension (add): A₂ = (1/2)π(5²) = 12.5π ≈ 39.27 m²

Shallow area to exclude (subtract): A₃ = 4 × 3 = 12 m²

Total: A = 200 + 39.27 - 12 = 227.27 m²

Answer: About 227 m²

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Tips for Success

Tip 1: Sketch and label

• Draw the figure neatly
• Label all dimensions
• Mark where you're splitting shapes

Tip 2: Look for patterns

• Can you make rectangles?
• Do you see triangles?
• Are there circles or parts of circles?

Tip 3: Choose your method

• Addition: When adding sections makes sense
• Subtraction: When removing a piece is easier
• Sometimes you can use either method!

Tip 4: Track units

• Keep units consistent (all feet or all meters)
• Final answer is in square units (ft², m², etc.)

Tip 5: Use formulas correctly

• Triangle: Don't forget the 1/2!
• Circle: π ≈ 3.14 or use calculator's π button
• Check which dimension is base vs. height

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Common Mistakes to Avoid

❌ Mistake 1: Counting overlapping areas twice

• Make sure pieces don't overlap!
• Each part of the figure should be counted exactly once

❌ Mistake 2: Using wrong dimensions

• Base and height must be perpendicular
• For circles, make sure you have radius (not diameter!)

❌ Mistake 3: Forgetting to add/subtract all pieces

• Count all sections of the composite figure

❌ Mistake 4: Unit errors

• Mixing feet and inches
• Forgetting to convert
• Writing final answer without square units

❌ Mistake 5: Calculation errors with π

• Semicircle is HALF the circle, don't forget the 1/2
• Use consistent value for π (3.14 or calculator)

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Practice Strategy

Step 1: Identify

• What basic shapes make up this figure?

Step 2: Decide on method

• Addition (add pieces) or subtraction (remove pieces)?

Step 3: Find missing dimensions

• Do you need to calculate any lengths?

Step 4: Calculate each area

• Use correct formula for each shape
• Show your work!

Step 5: Combine

• Add or subtract as needed
• Include units!

Step 6: Check reasonableness

• Does your answer make sense?
• Is it bigger/smaller than you expected?

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Different Ways to Break Up the Same Figure

Example: An L-shape can be split multiple ways:

Method 1: Two rectangles (horizontal cut) Method 2: Two rectangles (vertical cut) Method 3: Large rectangle minus missing corner

All methods give the SAME ANSWER! Choose whichever is easiest for you.

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Quick Reference

Addition Method: Total Area = A₁ + A₂ + A₃ + ...

Subtraction Method: Total Area = Large Area - Removed Area

Key Formulas:

• Rectangle: lw
• Triangle: (1/2)bh
• Circle: πr²
• Semicircle: (1/2)πr²

Remember:

• Break complex into simple
• Use appropriate formulas
• Add or subtract as needed
• Keep track of units!

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Summary

Composite figures are made up of multiple basic shapes.

Strategy:

1. Break into simpler shapes (rectangles, triangles, circles, etc.)
2. Find area of each piece using appropriate formula
3. Add areas together (or subtract if removing a piece)

Two Main Methods:

• Addition: Add all the pieces
• Subtraction: Large shape minus removed piece

Applications:

• Floor plans and architecture
• Landscaping and gardening
• Painting and flooring
• Manufacturing and design

Understanding composite figures is essential for real-world problem solving, construction, design, and advanced geometry!