Compound Shapes and Parts of Shapes (Copy)
Cheat Sheet: Compound Shapes and Solids – Perimeter, Area, Surface Area, Volume (IGCSE Mathematics 0580 – CORE)
Topic: Compound and Partial Shapes/Solids
1. Compound Shapes (Perimeter and Area)
- Compound shape = A shape made from two or more simple shapes combined (e.g., rectangle + semicircle, two rectangles joined together).
- Perimeter = Add all outer sides carefully.
- If a curve (like semicircle), use the appropriate part of circumference.
- Area =
- Break into simple shapes.
- Find area of each part separately.
- Add or subtract as needed to get total area.
Important:
- Perimeter of curved shapes: use fractions of full circle circumference.
- Straight and curved parts must be measured separately.
2. Parts of Shapes (Area and Perimeter)
- Half a circle (semicircle) area = (1/2)πr²
- Quarter of a circle area = (1/4)πr²
- Three-quarters of a circle area = (3/4)πr²
- Semicircle perimeter = (1/2 × circumference) + diameter
3. Compound Solids (Surface Area and Volume)
- Compound solid = A solid made up of more than one simple solid (e.g., cylinder + hemisphere).
- Surface Area:
- Add surface areas of each visible part.
- Do not double-count hidden or joined areas.
- Volume:
- Add volumes of each solid if they are combined.
- Subtract volumes if part is hollow or removed.
4. Parts of Solids (Surface Area and Volume)
- Half a sphere volume = (1/2) × (4/3)πr³ = (2/3)πr³
- Half a sphere surface area = (1/2 × 4πr²) + πr² = 3πr²
(half of surface + base circle) - Quarter sphere volume = (1/4) × (4/3)πr³ = (1/3)πr³
5. Key Formulas Reminder
| Shape/Solid | Area/Surface Area Formula | Volume Formula |
|---|---|---|
| Circle | Area = πr² | — |
| Rectangle | Area = length × width | — |
| Triangle | Area = (1/2) × base × height | — |
| Cylinder | Surface Area = 2πrh + 2πr² | Volume = πr²h |
| Sphere | Surface Area = 4πr² | Volume = (4/3)πr³ |
| Cone | Surface Area = πrl + πr² | Volume = (1/3)πr²h |
| Hemisphere (Half Sphere) | Surface Area = 3πr² | Volume = (2/3)πr³ |
6. Important Practical Tips
- Always check if the shape/solid is complete or only a part (e.g., half, quarter).
- Use π properly: leave in terms of π if asked, otherwise calculate π ≈ 3.1416.
- Label parts clearly when solving compound problems.
- Watch units carefully — keep them consistent.
This cheat sheet fully covers CORE-level methods for solving perimeter, area, surface area, and volume problems for compound and partial shapes and solids.