Compound shapes and parts of shapes
Compound shapes
- A compound shape is made by joining or cutting simple shapes.
- Split it into parts you know, then add or subtract the areas (or volumes).
- Worked example: a rectangle $8 \times 5$ with a semicircle (diameter $5$) on one end.
- Area $= 40 + \tfrac12 \pi (2.5)^2 = 40 + 3.125\pi \approx 49.8\ \text{cm}^2$.
Practice
To find the area of a rectangle with a semicircle on one end, you:
The shape is the rectangle plus the semicircle, so add the two areas.
Practice
A compound shape is a rectangle 8 cm by 5 cm plus a semicircle on the end. What is the rectangle part of the area (cm²)?
Rectangle area = 8 × 5 = 40 cm² (then add the semicircle).
Parts of solids
- Take the right fraction of a whole solid.
- A hemisphere (half a sphere): volume $= \tfrac12 \times \tfrac43 \pi r^3 = \tfrac23 \pi r^3$.
- A frustum is a cone with the top cut off — subtract the small top cone from the whole cone.
Practice
The volume of a hemisphere of radius r is:
Half of (4/3)πr³ is (2/3)πr³.
You've got it
Key idea
- split a compound shape into known parts, then add or subtract
- hemisphere volume $= \tfrac23 \pi r^3$
- a frustum = whole cone minus the small top cone