Solids: Surface Areas and Volumes
What the exam tests
Solids questions test curved and total surface areas and volumes of cylinders, cones, spheres and hemispheres. Kerala SSLC problems often combine solids, convert units, or compare volumes after a scale change.
Core rules
- Cylinder: curved area 2πrh; total area 2πr(r+h); volume πr²h.
- Cone: slant height l=√(r²+h²); curved area πrl; total area πr(l+r); volume (1/3)πr²h.
- Sphere: surface area 4πr²; volume (4/3)πr³.
- Hemisphere: curved area 2πr²; total area 3πr²; volume (2/3)πr³.
- Capacity is volume in litres: 1000 cm³=1 L.
- Areas use square units; volumes use cubic units.
Worked example
A cone has radius 7 cm and height 24 cm. Its slant height is l=√(7²+24²)=√(49+576)=√625=25 cm. Curved surface area =π×7×25=175π cm². Volume =(1/3)π×7²×24=392π cm³. If π=22/7, these become 550 cm² and 1232 cm³.
Traps
The vertical height and slant height are different; use Pythagoras to find l. Total surface area of a cone includes the circular base; curved surface area does not. A hemisphere’s total surface area includes the flat circular face, so it is 3πr². Do not confuse diameter with radius. When scaling a solid by factor k, areas scale by k² and volumes by k³.