Surface Areas and Volumes
Surface Areas and Volumes tests mensuration of cylinders, cones, spheres, hemispheres and combinations. The exam emphasises choosing curved or total surface area and using the correct volume formula.
Core rules
- Cylinder: CSA=2πrh; TSA=2πr(r+h); volume=πr²h.
- Cone: l=√(r²+h²); CSA=πrl; TSA=πr(l+r); volume=(1/3)πr²h.
- Sphere: surface area=4πr²; volume=(4/3)πr³.
- Hemisphere: CSA=2πr²; TSA=3πr²; volume=(2/3)πr³.
- For a combination, add exposed surfaces only and add volumes of joined solids.
- Frustum: volume=(1/3)πh(r1²+r1r2+r2²), if included in your syllabus.
Worked example
A cone has r=3 cm and h=4 cm. Its slant height l=√(3²+4²)=5 cm. CSA=π×3×5=15π cm². Volume=(1/3)π×3²×4=12π cm³.
Traps
Slant height is not the vertical height; compute l before using CSA or TSA. “Curved” excludes the base(s), while “total” includes them. When a solid is hollowed or joined, do not add hidden surfaces. Keep all dimensions in the same unit before calculating. For a hemisphere, TSA is 3πr², not 2πr².