Surface area and volume of solids
Volume of solids
| Solid | Volume |
|---|---|
| cuboid | length $\times$ width $\times$ height |
| prism | cross-section area $\times$ length |
| cylinder | $\pi r^2 h$ |
| pyramid | $\tfrac13 \times$ base area $\times h$ |
| cone | $\tfrac13 \pi r^2 h$ |
| sphere | $\tfrac43 \pi r^3$ |
Practice
A cylinder has radius 5 cm and height 10 cm. Its volume is kπ cm³. What is k?
Volume = πr²h = π(25)(10) = 250π, so k = 250.
Practice
The volume of a sphere of radius r is:
Sphere volume = (4/3)πr³; 4πr² is its surface area.
Surface area
- Add the area of every outside face.
- Cylinder: curved $2\pi r h$ plus two ends $2\pi r^2$. Sphere: $4\pi r^2$.
- Worked example: cylinder $r = 5$, $h = 10$ → volume $250\pi\ \text{cm}^3$; surface $2\pi(5)(10) + 2\pi(5)^2 = 100\pi + 50\pi = 150\pi\ \text{cm}^2$.
Practice
The same cylinder (r = 5, h = 10) has total surface area kπ cm². What is k?
2π(5)(10) + 2π(5²) = 100π + 50π = 150π, so k = 150.
You've got it
Key idea
- cylinder volume $\pi r^2 h$; sphere volume $\tfrac43\pi r^3$; cone volume $\tfrac13 \pi r^2 h$
- cylinder surface $= 2\pi r h + 2\pi r^2$; sphere surface $= 4\pi r^2$
- $r = 5$, $h = 10$ cylinder: volume $250\pi$, surface $150\pi$