Compound Shapes and Parts of Shapes
5.5 Compound shapes and parts of shapes – Cheat Sheet
Perimeter and area of compound shapes
- Break the shape into known simple shapes (rectangles, triangles, circles).
- Find the area of each part, then add or subtract as needed.
- Perimeter = sum of the outer boundary lengths (do not include internal edges).
Surface area and volume of compound solids
- Split the solid into known solids (cuboids, cylinders, cones, spheres, etc.).
- Calculate the surface area or volume of each part.
- Add or subtract as required.
- When finding surface area, remove hidden faces between joined solids.
Frustums
- A frustum is made by cutting a cone or pyramid parallel to its base.
Volume formula:
V = (1/3)πh(r₁² + r₂² + r₁r₂)
Curved surface area formula:
CSA = π(r₁ + r₂) × l
Slant height:
l = √((r₁ − r₂)² + h²)
Where:
- r₁ = radius of larger base
- r₂ = radius of smaller base
- h = vertical height
- l = slant height
Key points
- Always convert all lengths to the same unit before calculating.
- Use π as given in the question (or 3.142 if not specified).
- Area units = unit², Volume units = unit³.
Examples
- Area of rectangle (8 × 5 cm) + semicircle (r = 4 cm):
Area = (8 × 5) + ½π × 4²
= 40 + 8π cm² - Volume of frustum, r₁ = 6 cm, r₂ = 4 cm, h = 10 cm:
V = (1/3)π × 10 × (36 + 16 + 24)
= (1/3)π × 10 × 76
= 760π/3 cm³