Mensuration: Areas and Volumes of Solids
Mensuration: Areas and Volumes of Solids tests formula selection for cylinders, cones, spheres, hemispheres and combinations. The exam often hides the required surface-curved, total, or exposed-inside a real object.
Core rules
- Cylinder: curved surface 2πrh; total surface 2πr(h+r); volume πr²h.
- Cone: slant height l=√(r²+h²); curved surface πrl; total πr(l+r); volume (1/3)πr²h.
- Sphere: surface 4πr²; volume (4/3)πr³. Hemisphere: curved surface 2πr²; total 3πr²; volume (2/3)πr³.
- Frustum: volume (1/3)πh(r₁²+r₁r₂+r₂²); curved surface πl(r₁+r₂), with l=√[h²+(r₁−r₂)²].
- Composite solids: add volumes; for surface area, count only exposed surfaces.
Worked example
A cone has r=7 cm and h=24 cm. Then l=√(49+576)=25 cm. Curved surface=π×7×25=175π cm²; volume=(1/3)π×49×24=392π cm³.
Traps
Do not use height in place of slant height for a cone’s curved surface. For a closed hemisphere, total surface includes the flat circular base; for an open bowl it does not. Keep all lengths in the same unit before substituting. When solids are joined, hidden faces are not part of the external surface.