Solved Exercises on Area of Composite Shapes in Grade 4

Master area calculation for composite shapes: rectangles, squares, triangles, and L-shapes through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Rectangle + Rectangle
Exercise 1
Find the area of this composite shape:
A large rectangle (8 units × 5 units) with a smaller rectangle (3 units × 2 units) attached to its right side.
Definition:

Composite shape: A shape made up of two or more simpler shapes like rectangles, squares, or triangles.

Area calculation method:
  1. Break the composite shape into simpler shapes (rectangles)
  2. Calculate the area of each simple shape
  3. Add all the areas together
Large Rectangle
8×5=40
Small Rectangle
3×2=6
Total Area
40+6=46
Step 1: Break into simple shapes

Shape consists of: Large rectangle (8×5) and Small rectangle (3×2)

Step 2: Calculate each area

Large rectangle area: 8 × 5 = 40 square units

Small rectangle area: 3 × 2 = 6 square units

Step 3: Add the areas

Total area = 40 + 6 = 46 square units

Total area = 46 square units
Final answer:

The area of the composite shape is 46 square units.

Applied rules:

• Area of rectangle: length × width

• Composite area: Sum of areas of all parts

• Breaking down: Divide complex shapes into simpler ones

2 L-Shaped Figure
Exercise 2
Find the area of this L-shaped figure:
A 6×4 rectangle with a 2×2 square removed from the bottom right corner.
Definition:

L-shaped figure: A composite shape formed by removing one rectangle from another, creating an L pattern.

Full Rectangle
6×4=24
Removed Part
2×2=4
L-Shape Area
24-4=20
Step 1: Find the area of the full rectangle

Full rectangle: 6 × 4 = 24 square units

Step 2: Find the area of the part being removed

Removed square: 2 × 2 = 4 square units

Step 3: Subtract to find the L-shape area

L-shape area = 24 - 4 = 20 square units

Total area = 20 square units
Final answer:

The area of the L-shaped figure is 20 square units.

Applied rules:

• Area of rectangle: length × width

• L-shape area: Full area - Removed area

• Subtraction method: Useful when a shape has a cut-out portion

3 Rectangle + Triangle
Exercise 3
Find the area of this composite shape:
A rectangle (6 units × 3 units) with a triangle (base 6 units, height 4 units) on top.
Definition:

Rectangle + Triangle: A composite shape combining rectangular and triangular areas.

Rectangle
6×3=18
Triangle
(6×4)/2=12
Total Area
18+12=30
Step 1: Identify the shapes

Shape consists of: Rectangle (6×3) and Triangle (base 6, height 4)

Step 2: Calculate rectangle area

Rectangle area: 6 × 3 = 18 square units

Step 3: Calculate triangle area

Triangle area: (base × height) ÷ 2 = (6 × 4) ÷ 2 = 12 square units

Step 4: Add the areas

Total area = 18 + 12 = 30 square units

Total area = 30 square units
Final answer:

The area of the composite shape is 30 square units.

Applied rules:

• Area of rectangle: length × width

• Area of triangle: (base × height) ÷ 2

• Composite area: Sum of areas of all parts

Solution: Exercises 4 to 5
4 Two Rectangles Overlapping
Exercise 4
Find the area of this composite shape:
A horizontal rectangle (7 units × 3 units) overlapping a vertical rectangle (2 units × 6 units) in the middle.
Definition:

Overlapping shapes: When calculating area, subtract the overlapping area to avoid counting it twice.

Horizontal Rectangle
7×3=21
Vertical Rectangle
2×6=12
Overlap Area
2×3=6
Total Area
21+12-6=27
Step 1: Calculate area of horizontal rectangle

7 × 3 = 21 square units

Step 2: Calculate area of vertical rectangle

2 × 6 = 12 square units

Step 3: Find the overlapping area

The overlap is 2 × 3 = 6 square units (where they intersect)

Step 4: Calculate total area

Total area = 21 + 12 - 6 = 27 square units

Total area = 27 square units
Final answer:

The area of the overlapping composite shape is 27 square units.

Applied rules:

• Area of rectangle: length × width

• Overlapping areas: Subtract the overlap once to avoid double counting

• Total area formula: Area(A) + Area(B) - Area(A∩B)

5 Complex Composite Shape
Exercise 5
Find the area of this complex shape:
A 5×5 square with a 2×3 rectangle attached to the right side and a 3×2 rectangle attached to the bottom.
Definition:

Complex composite shape: A shape made up of multiple rectangles arranged in different positions.

Main Square
5×5=25
Right Rectangle
2×3=6
Bottom Rectangle
3×2=6
Total Area
25+6+6=37
Step 1: Identify all shapes

Main square (5×5), Right rectangle (2×3), Bottom rectangle (3×2)

Step 2: Calculate each area

Main square: 5 × 5 = 25 square units

Right rectangle: 2 × 3 = 6 square units

Bottom rectangle: 3 × 2 = 6 square units

Step 3: Check for overlaps

No overlaps exist between the shapes

Step 4: Add all areas

Total area = 25 + 6 + 6 = 37 square units

Total area = 37 square units
Final answer:

The area of the complex composite shape is 37 square units.

Applied rules:

• Area of square: side × side

• Area of rectangle: length × width

• Non-overlapping shapes: Simply add all areas together

Rules and methods, laws,...
Area = Length × Width
Area of Rectangle
Rectangle
A = l × w
Length times Width
Square
A = s²
Side squared
Triangle
A = ½bh
Half base times height
⬜
Rectangle
A = length × width
🟦
Square
A = side × side
🔺
Triangle
A = ½ × base × height
🏗️
Composite
Add individual areas
Key Concept: Breaking Down Shapes

Every composite shape can be divided into simpler shapes whose areas we know how to calculate.

Common Mistake Alert

Don't forget to subtract overlapping areas when shapes overlap!

Complete Guide: Area of Composite Shapes
Total Area = A₁ + A₂ + ... + Aₙ
Adding Individual Areas
Key definitions:

Composite shape: A shape made up of two or more basic geometric shapes like rectangles, squares, triangles, or circles.

Area: The amount of space inside a 2D shape, measured in square units.

Decomposition: Breaking a complex shape into simpler shapes whose areas we can calculate.

Complete methodology:
  1. Look at the shape: Identify the different parts that make up the composite shape
  2. Label the parts: Give each simple shape a number or name
  3. Measure dimensions: Find the length and width of each simple shape
  4. Calculate each area: Use the correct formula for each shape
  5. Add the areas: Sum all the individual areas together
Tip 1: Draw lines to separate the composite shape into rectangles, squares, or triangles.
Tip 2: Write down the area of each simple shape before adding them together.
Tip 3: Count the unit squares in a grid if you're not sure about your calculation.
Tip 4: Always include the word "square units" in your final answer.
Common errors: Forgetting to add all parts, miscalculating dimensions, confusing area with perimeter.
Exam preparation: Practice decomposing shapes, memorize area formulas, check your work by counting squares.
Formulas to know by heart:

• Area of rectangle: A = length × width

• Area of square: A = side × side (or side²)

• Area of triangle: A = ½ × base × height

• Area of composite shape: Add all individual areas

• When shapes overlap: Add areas and subtract the overlap

Questions & Answers

Question: I don't understand how to break down a complex shape into simpler parts. Can you explain how to identify the different rectangles in a composite shape?

Answer: Great question! Here's how to break down composite shapes:

  • Look for straight edges: Composite shapes are usually made of rectangles, squares, and triangles with straight sides.
  • Find the corners: Look for where lines meet at right angles (90°).
  • Trace the outlines: Imagine drawing separate boxes around each distinct rectangular part.
  • Check for L-shapes: These are often made of two rectangles joined together.

For example, an L-shape can be seen as either:

  • A large rectangle with a smaller rectangle removed, OR
  • Two rectangles joined together
Both approaches work! Just pick the one that seems easier to calculate.

Question: When I calculate the area of a shape with a part cut out (like a rectangle with a square hole), should I add or subtract the areas?

Answer: When a shape has a part cut out or removed, you SUBTRACT the area of the cut-out part from the area of the whole shape.

Think of it this way:

  • If you have a piece of paper (the whole shape) and you cut out a piece (the hole), you now have LESS paper.
  • So: Area of shape with hole = Area of whole shape - Area of hole
For example, if you have a 6×4 rectangle (area=24) with a 2×2 square cut out (area=4), the remaining area is 24 - 4 = 20 square units.

The key is to think: "Am I adding pieces together, or am I removing a piece?"

Question: How do I know if I've calculated the area correctly? Is there a way to check my answer?

Answer: Yes, there are several ways to check your area calculations:

  • Count the squares: If your shape is on grid paper, count the unit squares inside the shape.
  • Re-break the shape differently: Try dividing the shape in a different way and see if you get the same total.
  • Estimate: Does your answer make sense? A small shape shouldn't have a huge area.
  • Check your arithmetic: Go back and double-check your multiplication and addition.

For example, if you calculated an area of 30 square units for a shape that looks like it covers about 30 squares on grid paper, your answer is likely correct!

Always ask yourself: "Does this answer seem reasonable for the size of the shape?"