Solved Exercises on Area of Composite Figures in Grade 7

Master area of composite figures: rectangles, triangles, circles, and complex shapes through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Rectangle and Triangle
Exercise 1
A composite figure consists of a rectangle with dimensions 10 cm by 6 cm and a right triangle on top with base 10 cm and height 4 cm. Find the total area.
Definition:

Composite Figure: A shape made up of two or more simple geometric figures like rectangles, triangles, circles, etc.

Method for composite area:
  1. Break the figure into recognizable simple shapes
  2. Calculate the area of each shape separately
  3. Add all areas together
  4. Include the units (cm²)
Rectangle
10×6 = 60 cm²
Triangle
½(10)(4) = 20 cm²
Total
80 cm²
Step 1: Identify the simple shapes

Shape 1: Rectangle (10 cm × 6 cm)

Shape 2: Right Triangle (base = 10 cm, height = 4 cm)

Step 2: Calculate rectangle area

Area of rectangle = length × width

Area of rectangle = 10 × 6 = 60 cm²

Step 3: Calculate triangle area

Area of triangle = ½ × base × height

Area of triangle = ½ × 10 × 4 = 20 cm²

Step 4: Add the areas

Total area = Rectangle area + Triangle area

Total area = 60 + 20 = 80 cm²

Step 5: Verify the calculation

Check: 60 + 20 = 80 ✓

Total area = 80 cm²
Final answer:

The total area of the composite figure is 80 cm².

Applied rules:

• Area Addition: Total area = sum of individual areas

• Rectangle Area: A = length × width

• Triangle Area: A = ½bh

2 Rectangle and Semicircle
Exercise 2
A rectangle has dimensions 8 cm by 5 cm. A semicircle with diameter 8 cm is attached to one of the 8 cm sides. Find the total area of the figure.
Definition:

Semicircle: Half of a circle. Area of semicircle = ½πr², where r is the radius.

Rectangle
8×5 = 40 cm²
Semicircle
½π(4)² ≈ 25.1 cm²
Total
≈ 65.1 cm²
Step 1: Identify the shapes and dimensions

Rectangle: 8 cm × 5 cm

Semicircle: diameter = 8 cm, so radius = 4 cm

Step 2: Calculate rectangle area

Area of rectangle = 8 × 5 = 40 cm²

Step 3: Calculate semicircle area

Area of semicircle = ½πr²

Area of semicircle = ½ × π × 4²

Area of semicircle = ½ × π × 16 = 8π ≈ 25.1 cm²

Step 4: Add the areas

Total area = 40 + 25.1 = 65.1 cm²

Step 5: Express answer with π or decimal

Exact: (40 + 8π) cm²

Approximate: 65.1 cm²

Total area ≈ 65.1 cm²
Final answer:

The total area is (40 + 8π) cm² or approximately 65.1 cm².

Applied rules:

• Semicircle Area: A = ½πr²

• Radius from Diameter: r = d/2

• Area Addition: Sum of component areas

3 Area Subtraction
Exercise 3
A square with side length 12 cm has a circular hole with radius 3 cm cut out from the center. Find the area of the remaining figure.
Definition:

Area Subtraction: When a shape is cut out from another shape, subtract the area of the cut-out from the original area.

Square
12² = 144 cm²
Circle
π(3)² ≈ 28.3 cm²
Remaining
≈ 115.7 cm²
Step 1: Calculate the area of the square

Area of square = side²

Area of square = 12² = 144 cm²

Step 2: Calculate the area of the circle

Area of circle = πr²

Area of circle = π × 3² = 9π ≈ 28.3 cm²

Step 3: Subtract the circle area from the square area

Remaining area = Square area - Circle area

Remaining area = 144 - 9π ≈ 144 - 28.3 = 115.7 cm²

Step 4: Express in exact and approximate form

Exact: (144 - 9π) cm²

Approximate: 115.7 cm²

Step 5: Verify the result makes sense

Remaining area (115.7) < Original area (144) ✓

Remaining area ≈ 115.7 cm²
Final answer:

The area of the remaining figure is (144 - 9π) cm² or approximately 115.7 cm².

Applied rules:

• Area Subtraction: Remaining area = Original area - Cut-out area

• Square Area: A = s²

• Circle Area: A = πr²

Composite Area Properties and Methods
\(A_{total} = A_1 + A_2 + ... + A_n\)
Composite Area Addition
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Rectangle: A = length × width
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Triangle: A = ½bh
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Circle: A = πr²
Key definitions:

Composite Figure: A shape composed of two or more basic geometric shapes.

Area Addition: When shapes are joined together, add their areas.

Area Subtraction: When a shape is cut out from another, subtract the cut-out area.

Basic Shapes: Fundamental geometric figures like squares, rectangles, triangles, circles, etc.

Complete methodology:
  1. Examine the Figure: Identify all the component shapes
  2. Decide Operation: Determine if adding or subtracting areas
  3. Measure Components: Find dimensions of each basic shape
  4. Apply Formulas: Use appropriate area formulas for each shape
  5. Combine Areas: Add or subtract as determined
  6. State Result: Include proper units (cm², m², etc.)
Tip 1: Draw lines to separate the composite figure into basic shapes.
Tip 2: Label dimensions on your diagram to keep track of measurements.
Tip 3: Be careful with units—convert all measurements to the same unit.
Tip 4: Check if your final answer is reasonable compared to the given dimensions.
Formulas to know by heart:

• Rectangle: A = length × width

• Square: A = side²

• Triangle: A = ½bh

• Parallelogram: A = bh

• Trapezoid: A = ½(b₁ + b₂)h

• Circle: A = πr²

• Semicircle: A = ½πr²

• Quarter Circle: A = ¼πr²

Solution: Exercises 4 to 5
4 Complex L-Shape
Exercise 4
An L-shaped figure is formed by removing a 3 cm by 4 cm rectangle from a corner of a 10 cm by 8 cm rectangle. Find the area and perimeter of the resulting figure.
Definition:

L-Shape: A polygon formed by joining rectangles in an L configuration, often created by removing a rectangular section from a larger rectangle.

Large Rectangle
10×8 = 80 cm²
Small Rectangle
3×4 = 12 cm²
L-Shape Area
68 cm²
Step 1: Calculate the area of the original rectangle

Original area = 10 × 8 = 80 cm²

Step 2: Calculate the area of the removed rectangle

Removed area = 3 × 4 = 12 cm²

Step 3: Find the area of the L-shape

L-shape area = Original area - Removed area

L-shape area = 80 - 12 = 68 cm²

Step 4: Calculate the perimeter of the L-shape

Perimeter = 10 + 8 + 7 + 4 + 3 + 4 + 3 + 8 = 47 cm

(Add all outer edges of the L-shape)

Step 5: Verify the area calculation

We could alternatively divide the L-shape into two rectangles:

Top rectangle: 10 × 4 = 40 cm²

Bottom rectangle: 7 × 4 = 28 cm²

Total: 40 + 28 = 68 cm² ✓

Area = 68 cm², Perimeter = 47 cm
Final answer:

The area of the L-shaped figure is 68 cm² and the perimeter is 47 cm.

Applied rules:

• Area Subtraction: A_remaining = A_original - A_removed

• Perimeter Calculation: Sum of all outer sides

• Alternative Method: Break into rectangles and add areas

5 Circular Segments
Exercise 5
A square with side length 14 cm has four quarter circles with radius 3.5 cm, one centered at each corner. Find the area of the shaded region inside the square but outside the quarter circles.
Definition:

Quarter Circle: One-fourth of a circle. Four quarter circles can form a complete circle.

Square
14² = 196 cm²
Four Quarter Circles
4 × ¼π(3.5)² = π(3.5)² ≈ 38.5 cm²
Shaded Area
≈ 157.5 cm²
Step 1: Calculate the area of the square

Area of square = 14² = 196 cm²

Step 2: Calculate the total area of the quarter circles

Area of one quarter circle = ¼πr²

Area of one quarter circle = ¼π(3.5)² = ¼π(12.25) = 3.0625π cm²

Area of four quarter circles = 4 × 3.0625π = 12.25π ≈ 38.5 cm²

Step 3: Find the shaded area

Shaded area = Square area - Quarter circles area

Shaded area = 196 - 12.25π ≈ 196 - 38.5 = 157.5 cm²

Step 4: Alternative approach

Four quarter circles form one complete circle

Area of complete circle = πr² = π(3.5)² = 12.25π ≈ 38.5 cm²

This confirms our previous calculation

Step 5: Express exact and approximate answers

Exact: (196 - 12.25π) cm²

Approximate: 157.5 cm²

Shaded area ≈ 157.5 cm²
Final answer:

The area of the shaded region is (196 - 12.25π) cm² or approximately 157.5 cm².

Applied rules:

• Area Subtraction: Shaded area = Total area - Unshaded area

• Quarter Circle Area: A = ¼πr²

• Circle Formation: Four quarter circles = one full circle

Complete Guide: Area of Composite Figures
\(A_{composite} = \sum A_{individual} \text{ or } A_{original} - A_{removed}\)
Composite Area Formula
Key definitions:

Composite Figure: A shape formed by combining two or more simple geometric figures.

Area Addition: The process of finding the total area by adding the areas of individual components.

Area Subtraction: The process of finding the remaining area by subtracting the area of removed parts from the original area.

Basic Shapes: Fundamental geometric figures including rectangles, squares, triangles, circles, and semicircles.

Complete methodology:
  1. Visualize the Figure: Identify how the composite figure is constructed from basic shapes
  2. Determine the Approach: Decide whether to add or subtract areas
  3. Identify Component Shapes: Name and measure each simple geometric figure
  4. Apply Area Formulas: Calculate the area of each component
  5. Combine Areas: Add or subtract according to the figure's construction
  6. Verify Solution: Check that the answer is reasonable and in proper units
Tip 1: Always sketch the figure and label dimensions to visualize the problem.
Tip 2: Look for ways to reorganize the figure into simpler shapes.
Tip 3: When subtracting, ensure you're subtracting the correct portion.
Tip 4: Express answers in both exact form (with π) and approximate decimal form.
Formulas to know by heart:

• Rectangle: A = length × width

• Square: A = side²

• Triangle: A = ½bh

• Parallelogram: A = bh

• Trapezoid: A = ½(b₁ + b₂)h

• Circle: A = πr²

• Semicircle: A = ½πr²

• Quarter Circle: A = ¼πr²

Questions & Answers

Question: How do I know when to add areas and when to subtract areas in composite figures?

Answer: The key is to understand how the figure is constructed:

Add Areas When:

  • Shapes are joined together to form a larger figure
  • You have distinct sections that make up the whole
  • Example: Rectangle with triangle on top

Subtract Areas When:

  • A shape is cut out from another shape
  • You need the area of the remaining part
  • Example: Square with circular hole in center

Ask yourself: "Am I putting shapes together or taking a piece away?" If pieces are added together, add areas. If a piece is removed, subtract areas.

Question: Can I break down a composite figure in different ways? Will I get the same answer?

Answer: Yes, you can decompose a composite figure in multiple valid ways, and you should get the same total area:

Example - L-Shape:

  • Method 1: Large rectangle minus small rectangle
  • Method 2: Two rectangles added together
  • Both methods yield the same result

The decomposition method you choose may affect the complexity of the calculation, but the final area should always be the same. This is a good way to verify your answer!

Question: How do I handle composite figures that include parts of circles like semicircles or quarter circles?

Answer: For circular parts in composite figures:

Area Formulas:

  • Semicircle: A = ½πr²
  • Quarter Circle: A = ¼πr²
  • Three-quarter Circle: A = ¾πr²
  • Sector: A = (central angle/360°) × πr²

Remember that the radius is always the distance from the center to the edge. Treat these curved shapes just like any other component of the composite figure.