Solved Exercises on Area of Composite Shapes with Circles in Grade 4

Master area of composite shapes with circles: semicircles, quarter circles, circles combined with rectangles through these 3 detailed exercises.

Solution: Exercises 1 to 3
1 Rectangle with Semicircle
Exercise 1
Find the area of this composite shape:
A rectangle (8 cm × 5 cm) with a semicircle (radius 4 cm) on top
Definition:

Semicircle: Half of a circle with area = (π × r²) ÷ 2

Method:
  1. Calculate the area of the rectangle
  2. Calculate the area of the semicircle
  3. Add the two areas together
Step 1: Calculate area of rectangle

Area of rectangle = length × width = 8 cm × 5 cm = 40 cm²

Step 2: Calculate area of semicircle

Area of circle = π × r² = 3.14 × 4² = 3.14 × 16 = 50.24 cm²

Area of semicircle = 50.24 ÷ 2 = 25.12 cm²

Step 3: Add the areas

Total area = 40 cm² + 25.12 cm² = 65.12 cm²

Total area ≈ 65.12 cm²
Final answer:

The area of the composite shape is approximately 65.12 cm²

Applied rules:

• Area of rectangle: length × width

• Area of circle: π × r²

• Area of semicircle: (π × r²) ÷ 2

2 Square with Circle Cut Out
Exercise 2
Find the area of this composite shape:
A square (10 cm × 10 cm) with a circle (radius 3 cm) cut out
Definition:

Circle area: π × r² (where π ≈ 3.14)

Step 1: Calculate area of square

Area of square = side × side = 10 cm × 10 cm = 100 cm²

Step 2: Calculate area of circle

Area of circle = π × r² = 3.14 × 3² = 3.14 × 9 = 28.26 cm²

Step 3: Subtract the circle area

Total area = 100 cm² - 28.26 cm² = 71.74 cm²

Total area ≈ 71.74 cm²
Final answer:

The area of the composite shape is approximately 71.74 cm²

Applied rules:

• Area of square: side × side

• Area of circle: π × r²

• Subtracted area: Main area - cut-out area

3 Rectangle with Quarter Circle
Exercise 3
Find the area of this composite shape:
A rectangle (6 cm × 4 cm) with a quarter circle (radius 2 cm) attached
Definition:

Quarter circle: One-fourth of a circle with area = (π × r²) ÷ 4

Step 1: Calculate area of rectangle

Area of rectangle = length × width = 6 cm × 4 cm = 24 cm²

Step 2: Calculate area of quarter circle

Area of full circle = π × r² = 3.14 × 2² = 3.14 × 4 = 12.56 cm²

Area of quarter circle = 12.56 ÷ 4 = 3.14 cm²

Step 3: Add the areas

Total area = 24 cm² + 3.14 cm² = 27.14 cm²

Total area ≈ 27.14 cm²
Final answer:

The area of the composite shape is approximately 27.14 cm²

Applied rules:

• Area of rectangle: length × width

• Area of quarter circle: (π × r²) ÷ 4

• Composite area: Sum of individual areas

Area Formulas and Methods
Area = π × r²
Circle Area
Area = (π × r²) ÷ 2
Semicircle Area
Area = (π × r²) ÷ 4
Quarter Circle Area
Key definitions:

Circle: A round shape where all points are equidistant from the center

Radius: Distance from center to edge of circle

Pi (π): Mathematical constant ≈ 3.14

Semicircle: Half of a circle

Quarter circle: One-fourth of a circle

Complete methodology:
  1. Identify shapes: Recognize circles, semicircles, or quarter circles in the composite figure
  2. Measure dimensions: Find the radius of circular parts
  3. Calculate areas: Use correct formulas for each shape
  4. Combine areas: Add or subtract as needed
  5. Round appropriately: Round to required decimal places
Tip 1: Remember π ≈ 3.14 for calculations in grade 4.
Tip 2: Semicircle area is half of a full circle's area.
Tip 3: Quarter circle area is one-fourth of a full circle's area.
Tip 4: When circles are cut out, subtract their area from the main shape.
Common mistakes: Forgetting π in calculations, misidentifying circle parts, incorrect radius measurement.
Success strategies: Draw dashed lines to separate shapes, double-check radius, use calculator for π.
Essential formulas:

• Circle area: π × r²

• Semicircle area: (π × r²) ÷ 2

• Quarter circle area: (π × r²) ÷ 4

• Composite area: sum or difference of parts

Detailed Summary
Concepts and Keywords:

Circle: A perfectly round shape where every point on the edge is the same distance from the center.

Radius: The distance from the center of a circle to any point on its edge.

Diameter: The distance across a circle through its center (equal to 2 × radius).

Pi (π): A special number that represents the ratio of a circle's circumference to its diameter (approximately 3.14).

Semicircle: Exactly half of a circle, formed by cutting a circle along its diameter.

Quarter circle: One-fourth of a circle, formed by cutting a circle into four equal parts.

Core Rules and Principles:
  1. Circle area rule: Area = π × radius² (always multiply π by the radius squared)
  2. Part-circle rule: Semicircle = full circle ÷ 2; Quarter circle = full circle ÷ 4
  3. Combination rule: Add areas when shapes are joined; subtract when shapes are cut out
  4. Measurement rule: Always measure the radius (not diameter) for area calculations
Examples progression: Simple circle → Semicircle → Quarter circle → Complex combinations.
Common pitfalls: Confusing radius with diameter, forgetting π in formulas, miscounting circle portions.
Memorization tips: Remember "Pi are squared" for circle area; "Half Pi r squared" for semicircle.
Visual Guide to Composite Shapes with Circles
Circular Shapes
⭕
Full Circle

Complete round shape

Area = π × r²

Semicircle

Half of a circle

Area = (π × r²) ÷ 2

Quarter Circle

One-fourth of a circle

Area = (π × r²) ÷ 4

Composite Examples
🧩
Rectangle + Semicircle

Areas are added

Rectangle + Semicircle

Square - Circle

Areas are subtracted

Square - Circle

Problem-Solving Steps
📋
How to Solve
1
Identify Shapes
2
Find Radius
3
Calculate Areas
4
Combine Areas
Formula Reference

Full Circle: π × r²

Semicircle: (π × r²) ÷ 2

Quarter Circle: (π × r²) ÷ 4

Where π ≈ 3.14

Practice Makes Perfect!
Q&A: Common Questions

Question: How do I find the radius of a circle in a composite shape if it's not labeled?

Answer: Great question! Here are ways to find the radius:

  • Look for clues: If the circle fits perfectly inside a square, the radius might be half the side of the square
  • Measure diameter: If you can see the full width of the circle, divide by 2 to get the radius
  • Compare to known lengths: The radius might match other dimensions in the shape
  • Count grid squares: If on grid paper, count squares from center to edge

Always look for relationships between the circle and other parts of the composite shape. The radius often relates to other measurements in the figure.

Question: Why do we use 3.14 for π? Is that exact?

Answer: Pi (π) is a special number that represents the relationship between a circle's circumference and diameter. It's an irrational number, which means it goes on forever without repeating:

π = 3.14159265358979323846...

For grade 4 math, we use 3.14 as an approximation because it's close enough for our calculations and easier to work with. In more advanced math, you might use more decimal places like 3.14159, but 3.14 works perfectly for these problems!

Question: How can I tell if a curved shape is a semicircle or a quarter circle?

Answer: Look at the angle of the curve and how much of the circle is shown:

  • Semicircle: Looks like a half-moon shape - it's exactly half of a full circle (180°)
  • Quarter circle: Looks like a corner piece - it's one-fourth of a full circle (90°)
  • Visual check: A semicircle has a straight line across the bottom; a quarter circle has two straight edges forming a corner

Think of it like pizza slices: a semicircle is like cutting a pizza in half, while a quarter circle is like cutting it into quarters. The angle at the center tells you how much of the circle you have!