Calculations Involving Compound Shapes
Key Concepts and Strategies
Approach to Solving Compound Shape Problems
- Decompose the shape into simpler geometric components
- Identify all known measurements and relationships
- Calculate areas/volumes of individual components
- Combine or subtract components as needed
- Convert units if necessary for consistency
Common Compound Shape Formulas
Circle in Square
Shaded Area: A = a² - π(a/2)²
Where a is side length of square
Sector of Circle
Area: A = (θ/360) × πr²
Arc Length: L = (θ/360) × 2πr
Where θ is angle in degrees, r is radius
Composite Solids
Volume: Sum of individual volumes
Surface Area: Sum of exposed surfaces
Watch for overlapping areas
Shaded Regions
Area: Often A = A₁ - A₂
Where A₁ is larger area, A₂ is subtracted area
Example: Circle area minus inscribed shape area
Worked Examples
Example 1: Circle in Square
Problem: A circle is inscribed in a square with area 81 cm². Calculate the shaded area (area outside the circle but inside the square).
Solution:
Square side length = √81 = 9 cm
Circle radius = 9/2 = 4.5 cm
Circle area = π × (4.5)² ≈ 3.1416 × 20.25 ≈ 63.62 cm²
Shaded area = Square area - Circle area = 81 - 63.62 = 17.38 cm²
Example 2: Sector and Triangle
Problem: A shape consists of a quarter-circle (radius 5 cm) and a right-angled triangle (base 6 cm). Calculate the total area.
Solution:
Quarter-circle area = ¼ × π × 5² = ¼ × π × 25 ≈ 19.63 cm²
Triangle area = ½ × base × height = ½ × 6 × 5 = 15 cm²
Total area = 19.63 + 15 = 34.63 cm²
Example 3: Cone and Hemisphere
Problem: A toy consists of a cone (radius 4 cm, height 9 cm) on top of a hemisphere (radius 4 cm). Calculate the volume.
Solution:
Cone volume = ⅓ × π × 4² × 9 = ⅓ × π × 16 × 9 = 48π ≈ 150.80 cm³
Hemisphere volume = ½ × ⁴⁄₃ × π × 4³ = ½ × ⁴⁄₃ × π × 64 ≈ 134.04 cm³
Total volume = 150.80 + 134.04 = 284.84 cm³
Problem-Solving Strategies
1. Decomposition
Break complex shapes into simpler components (rectangles, triangles, circles).
2. Identify Relationships
Look for geometric relationships (tangents, right angles, symmetries).
3. Sequential Calculation
Calculate areas/volumes of components in logical order.
4. Check Units
Ensure all measurements use consistent units before calculating.
Real-World Application: Percentage Area
A square has side length k cm. A sector is drawn with center at one vertex, passing through the midpoint of an adjacent side. Calculate the percentage of the square that is shaded.
Solution Approach:
1. Calculate area of square: k²
2. Determine radius of sector: k (distance from vertex to midpoint)
3. Calculate area of sector: (θ/360) × π × k²
4. Calculate shaded area: Square area - Sector area
5. Calculate percentage: (Shaded area / Square area) × 100%
Practice Quiz
Question 1
A circle with diameter 7 cm touches two sides of a parallelogram with base 12 cm. Calculate the shaded area.
Question 2
A square has vertices on a circle with radius 6 cm. Calculate the area between the circle and the square.
Question 3
A solid consists of a cone and a hemisphere, both with radius 6.2 cm. The total surface area is 600 cm². Calculate the slant height of the cone.
Question 4
A regular octagon has sides of length 6 cm. Calculate the area of the circle that passes through its vertices.
Question 5
A cone contains salt to a depth of 4.5 cm. The cone has radius 1.75 cm and height 6 cm. Calculate the volume of salt.
Quiz Answers
- Area of parallelogram = base × height = 12 × 7 = 84 cm². Area of circle = π × (3.5)² ≈ 38.48 cm². Shaded area = 84 - 38.48 = 45.52 cm²
- Area of circle = π × 6² ≈ 113.10 cm². Square diagonal = 12 cm, so side = 12/√2 ≈ 8.49 cm. Square area ≈ 72.07 cm². Shaded area ≈ 113.10 - 72.07 = 41.03 cm²
- Hemisphere surface area = ½ × 4π × (6.2)² ≈ 241.90 cm². Cone curved surface area = 600 - 241.90 = 358.10 cm². Then l = (358.10)/(π × 6.2) ≈ 18.39 cm
- Central angle = 360°/8 = 45°. Using trigonometry, distance from center to vertex = 6/(2×sin(22.5°)) ≈ 7.84 cm. Circle area = π × (7.84)² ≈ 193.14 cm²
- Using similar triangles, radius at salt level = (4.5/6) × 1.75 = 1.3125 cm. Volume = ⅓ × π × (1.3125)² × 4.5 ≈ 8.14 cm³
Geometric Relationships Reference
| Relationship | Formula/Principle | Application |
|---|---|---|
| Circle in Square | Diameter = Side length | Area calculations |
| Square in Circle | Diagonal = Diameter | Finding side length |
| Similar Triangles | Ratio of sides preserved | Partial volumes/heights |
| Regular Polygons | Central angle = 360°/n | Area calculations |
Exam Tips
Draw Diagrams
Sketch the shape and label all known measurements.
Show Your Method
Clearly show each step of your calculation for partial credit.
Check Dimensions
Verify that your answer has the correct units (cm², cm³, etc.).
Estimate First
Make a rough estimate to check if your final answer is reasonable.
Common Mistake to Avoid
When calculating partial volumes in cones or pyramids, remember that the dimensions scale linearly with height, but the volume scales with the cube of the height ratio.