Area of Rectilinear Polygon - 3rd Grade Math

Master area calculation with visual learning: decomposing rectilinear polygons, measuring parts, and step-by-step solutions for 3rd grade students.

Rectilinear Polygon Concepts and Area
Area = Sum of rectangle areas
Area Formula
Key Definitions:

Rectilinear Polygon: A polygon made up of rectangles joined together

Polygon: A closed shape with straight sides

Decompose: To break a shape into smaller parts

Rectangle: A 4-sided shape with 4 right angles

Area: The amount of space inside a shape measured in square units

Partial Area: The area of a part of the polygon

Total Area: The combined area of all parts

Dimension: Length and width of a shape

Area Calculation Method:
  1. Identify the shape: Confirm it's a rectilinear polygon
  2. Break into rectangles: Draw lines to separate the shape into rectangles
  3. Measure dimensions: Find length and width of each rectangle
  4. Calculate areas: Find area of each rectangle using length × width
  5. Add partial areas: Sum all rectangle areas together
  6. Record result: Include "square units" in your answer
Tip 1: Look for right angles to identify where rectangles meet.
Tip 2: Draw lines to separate the shape into non-overlapping rectangles.
Tip 3: Always add the areas of all parts to get the total area.
Tip 4: Count squares inside each rectangle to verify your calculation.
Decomposition Strategy: Break complex shapes into simpler rectangles.
Area Principle: The area of a shape equals the sum of its parts.
Essential Rules:

• Rectilinear polygons can be decomposed into rectangles

• Area of each rectangle = Length × Width

• Total area = Sum of all rectangle areas

• Units must be squared in final answer

• All parts must be included in the sum

Exercise Solutions: 1 to 3
1 Basic L-Shape
Exercise 1
Find the area of an L-shaped polygon that can be divided into two rectangles: one 4 units by 2 units and another 2 units by 3 units.
Problem Analysis:

Shape: L-shaped rectilinear polygon

Components: Rectangle 1 (4×2), Rectangle 2 (2×3)

Method: Calculate area of each rectangle, then add

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Part 1
(4×2)
Part 2
(2×3)
Part 1
4 × 2 = 8
Part 2
2 × 3 = 6
Total
8 + 6 = 14
Step 1: Identify the rectangles

Rectangle 1: 4 units by 2 units

Rectangle 2: 2 units by 3 units

Step 2: Calculate area of first rectangle

Area of Rectangle 1 = 4 × 2 = 8 square units

Step 3: Calculate area of second rectangle

Area of Rectangle 2 = 2 × 3 = 6 square units

Step 4: Add the areas

Total Area = 8 + 6 = 14 square units

Step 5: Record the answer

Area = 14 square units

Area = 14 square units
Final answer:

The area of the L-shaped polygon is 14 square units.

Applied rules:

• Decomposition: Break shape into rectangles

• Area formula: Length × Width for each rectangle

• Sum of parts: Add all partial areas

2 T-Shape
Exercise 2
Find the area of a T-shaped polygon that can be divided into three rectangles: one 5 units by 2 units and two 1 unit by 3 units each.
Problem Analysis:

Shape: T-shaped rectilinear polygon

Components: Rectangle 1 (5×2), Rectangle 2 (1×3), Rectangle 3 (1×3)

Method: Calculate area of each rectangle, then add

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Top
(5×2)
Left
(1×3)
Center
(1×3)
Right
(1×3)
Top
5 × 2 = 10
Left
1 × 3 = 3
Center
1 × 3 = 3
Right
1 × 3 = 3
Total
10+3+3+3=19
Step 1: Identify the rectangles

Rectangle 1: 5 units by 2 units (top bar)

Rectangle 2: 1 unit by 3 units (left stem)

Rectangle 3: 1 unit by 3 units (center stem)

Rectangle 4: 1 unit by 3 units (right stem)

Step 2: Calculate area of each rectangle

Top: 5 × 2 = 10 square units

Left: 1 × 3 = 3 square units

Center: 1 × 3 = 3 square units

Right: 1 × 3 = 3 square units

Step 3: Add all areas

Total Area = 10 + 3 + 3 + 3 = 19 square units

Step 4: Record the answer

Area = 19 square units

Area = 19 square units
Final answer:

The area of the T-shaped polygon is 19 square units.

Applied rules:

• Decomposition: Break shape into rectangles

• Area formula: Length × Width for each rectangle

• Sum of parts: Add all partial areas

3 Complex Polygon
Exercise 3
Find the area of a rectilinear polygon that can be divided into four rectangles: one 3×2, one 2×4, one 1×3, and one 2×2.
Problem Analysis:

Shape: Complex rectilinear polygon

Components: Rectangle 1 (3×2), Rectangle 2 (2×4), Rectangle 3 (1×3), Rectangle 4 (2×2)

Method: Calculate area of each rectangle, then add

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R1
(3×2)
R2
(2×4)
R3
(1×3)
R4
(2×2)
R1
3×2=6
R2
2×4=8
R3
1×3=3
R4
2×2=4
Total
6+8+3+4=21
Step 1: Identify all rectangles

Rectangle 1: 3 units by 2 units

Rectangle 2: 2 units by 4 units

Rectangle 3: 1 unit by 3 units

Rectangle 4: 2 units by 2 units

Step 2: Calculate area of each rectangle

R1: 3 × 2 = 6 square units

R2: 2 × 4 = 8 square units

R3: 1 × 3 = 3 square units

R4: 2 × 2 = 4 square units

Step 3: Add all areas

Total Area = 6 + 8 + 3 + 4 = 21 square units

Step 4: Record the answer

Area = 21 square units

Area = 21 square units
Final answer:

The area of the rectilinear polygon is 21 square units.

Applied rules:

• Decomposition: Break shape into rectangles

• Area formula: Length × Width for each rectangle

• Sum of parts: Add all partial areas

Exercise Solutions: 4 to 5
4 Real-world Application
Exercise 4
A floor plan consists of a rectilinear polygon made up of two rectangles: a living room (8 feet by 6 feet) and a kitchen (4 feet by 5 feet). What is the total area of the floor plan?
Problem Analysis:

Context: Floor plan with living room and kitchen

Components: Living room (8×6), Kitchen (4×5)

Method: Calculate area of each room, then add

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Living Room
8 × 6 = 48
Kitchen
4 × 5 = 20
Total
48 + 20 = 68
Step 1: Identify the rectangles

Living Room: 8 feet by 6 feet

Kitchen: 4 feet by 5 feet

Step 2: Calculate area of living room

Area = 8 × 6 = 48 square feet

Step 3: Calculate area of kitchen

Area = 4 × 5 = 20 square feet

Step 4: Add the areas

Total Area = 48 + 20 = 68 square feet

Step 5: State the final answer

The total area of the floor plan is 68 square feet

Total area = 68 square feet
Final answer:

The total area of the floor plan is 68 square feet.

Applied rules:

• Real-world context: Area represents physical space

• Decomposition: Break complex shape into rectangles

• Area formula: Length × Width for each rectangle

5 Advanced Problem
Exercise 5
A rectilinear polygon garden consists of three rectangular sections: a vegetable patch (6 feet by 4 feet), a flower bed (3 feet by 5 feet), and a pathway (2 feet by 8 feet). If it costs $3 per square foot to landscape the garden, how much will it cost to landscape the entire garden?
Problem Analysis:

Components: Vegetable patch (6×4), Flower bed (3×5), Pathway (2×8)

Cost: $3 per square foot

Steps: Find total area, then multiply by cost per square foot

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Step 1
Area = (6×4)+(3×5)+(2×8)
Step 2
Area = 24+15+16=55
Step 3
Cost = 55×$3=$165
Step 1: Calculate area of vegetable patch

Area = 6 × 4 = 24 square feet

Step 2: Calculate area of flower bed

Area = 3 × 5 = 15 square feet

Step 3: Calculate area of pathway

Area = 2 × 8 = 16 square feet

Step 4: Calculate total area

Total Area = 24 + 15 + 16 = 55 square feet

Step 5: Calculate landscaping cost

Cost = 55 × $3 = $165

Landscaping cost = $165
Final answer:

It will cost $165 to landscape the entire garden.

Applied rules:

• Decomposition: Break complex shape into rectangles

• Area formula: Length × Width for each rectangle

• Cost calculation: Total area × Cost per unit area

Questions & Answers

Question: How do I know where to draw the lines to break a rectilinear polygon into rectangles?

Answer: Great question! Here are some strategies to find where to draw the lines:

  • Look for right angles: Draw lines at the corners where rectangles meet
  • Find edges that line up: If two sides are aligned, draw a line between them
  • Keep it simple: Try to create as few rectangles as possible
  • Make sure all shapes are rectangles: Each piece should have 4 right angles

Sometimes there are multiple ways to break a shape, but they all give the same total area. The key is to make sure all pieces are rectangles and nothing is missed or counted twice.

Question: How can I help my child practice area of rectilinear polygons?

Answer: Here are some fun and effective practice ideas:

  • Grid paper: Draw rectilinear polygons and have your child count squares
  • House plans: Use floor plans to practice with real-world examples
  • Building blocks: Create shapes with unit cubes to visualize area
  • Art projects: Cut shapes from construction paper and decompose them
  • Online tools: Use virtual geoboards or geometry apps

Start with simple L-shapes and gradually move to more complex polygons. The key is lots of visual practice!

Question: What if I accidentally count some squares twice when finding the area?

Answer: Good question! To avoid counting squares twice:

  • Label each rectangle: Give each part a different color or letter
  • Count systematically: Go row by row or column by column
  • Check for overlaps: Make sure rectangles don't share the same squares
  • Mark as you count: Put a dot in each square as you count it

If you're adding rectangle areas, make sure the rectangles don't overlap. If they do, you'll be counting those overlapping parts twice, which gives the wrong answer!