Mensuration: Solids and Combined Solids
Solid formulae
Cylinder: CSA=2πrh, TSA=2πr(r+h), V=πr²h. Cone: slant height l=√(r²+h²), CSA=πrl, TSA=πr(l+r), V=⅓πr²h. Sphere: surface area=4πr², V=4/3πr³. Hemisphere: CSA=2πr², TSA=3πr², V=2/3πr³. Frustum: l=√[h²+(r₁−r₂)²], CSA=πl(r₁+r₂), TSA=πl(r₁+r₂)+πr₁²+πr₂², V=⅓πh(r₁²+r₁r₂+r₂²).
Combined solids
When solids are joined, volumes add. For surface area, count only exposed surfaces; a joined circular face is removed from each solid. For a cylinder capped by a hemisphere, TSA=2πrh+2πr² (or πr² if the hemisphere replaces an open top, depending on the wording).
Worked example
A cylinder has r=7 cm and h=10 cm. V=πr²h=(22/7)×49×10=1540 cm³. Its CSA=2πrh=2×(22/7)×7×10=440 cm². TSA=2πr(r+h)=2×(22/7)×7×17=748 cm².
Traps
Using height instead of slant height for a cone’s CSA; forgetting to subtract hidden faces in combined solids; mixing radius and diameter; and leaving answers without cubic or square units. In a frustum, l is not h.