Chapter 12: Surface Areas & Volumes
Mastering combined solids: Cylinders, Cones, Hemispheres, and Cuboids.
📘 Combined Solids
Real-life objects like tents, capsules, and toys are usually combinations of two or more basic solids.
When joining two solids, the faces where they connect are hidden inside. Never add their Total Surface Areas (TSA) directly!
⚙️ Area vs Volume Logic
Sum of the Curved Surface Areas (CSA) of the visible parts + any exposed flat bases.
Total Volume of combinationVolume is simply the sum of the volumes of the individual constituents (unlike area, nothing disappears when combined).
| Solid Shape | Volume (V) | Curved Surface Area (CSA) | Total Surface Area (TSA) |
|---|---|---|---|
| Cuboid | l × b × h | 2h(l + b) | 2(lb + bh + hl) |
| Cube | a³ | 4a² | 6a² |
| Cylinder | πr²h | 2πrh | 2πr(r + h) |
| Cone |
13 πr²h
|
πrl | πr(r + l) |
| Sphere |
43 πr³
|
4πr² | 4πr² |
| Hemisphere |
23 πr³
|
2πr² | 3πr² |
🧩 Classic Combinations
Learn these standard structures to solve 90% of exam problems!
Cylinder + 2 Hemispheres
+ 2 × CSA(Hemisphere)
Cone + Hemisphere
+ CSA(Hemisphere)
Cone + Cylinder
+ CSA(Cylinder)
⭐ Board Exam Super-Hacks
The "Total Height" Trick
If a toy's total height is given, find the height of the cylindrical/conical part by subtracting the radius of the hemispherical part(s) from the total height!
Hollowing Out Solids
When a cavity is "hollowed out" (e.g., cutting a cone from a cylinder), the volume decreases, but the Surface Area INCREASES because a new inner surface is created!
⚠️ Common Exam Traps
Trap 1: Forget to calculate 'l'
For cones, the formula for CSA uses slant height 'l' (πrl), NOT perpendicular height 'h'. Always calculate l = √(r² + h²) first!
Trap 2: The "Surmounted" Cube
If a hemisphere sits on a cube, don't just add their areas. You must subtract the circular base of the hemisphere that is covering the top of the cube: TSA = 6a² + 2πr² - πr².