Chapter at a Glance

Every problem here is one of four types: basic solid (use a formula), combination surface area (add exposed curved surfaces), combination volume (add, or subtract a cavity), conversion (old volume = new volume), or frustum (its own three formulas). Identify the type first.

Base Formulas (Must Memorise)

Solid Curved SA Total SA Volume
Cuboid (l,b,h)(l,b,h) 2h(l+b)2h(l+b) 2(lb+bh+hl)2(lb+bh+hl) lbhlbh
Cube (a)(a) 4a24a^2 6a26a^2 a3a^3
Cylinder (r,h)(r,h) 2πrh2\pi rh 2πr(r+h)2\pi r(r+h) πr2h\pi r^2h
Cone (r,h,l)(r,h,l) πrl\pi rl πr(l+r)\pi r(l+r) 13πr2h\tfrac13\pi r^2h
Sphere (r)(r) 4πr24\pi r^2 4πr24\pi r^2 43πr3\tfrac43\pi r^3
Hemisphere (r)(r) 2πr22\pi r^2 3πr23\pi r^2 23πr3\tfrac23\pi r^3

Cone slant height: l=r2+h2l=\sqrt{r^2+h^2}. Cube diagonal a3a\sqrt3; cuboid diagonal l2+b2+h2\sqrt{l^2+b^2+h^2}.

Combination, Conversion and Frustum Rules

  • Surface area of a combination: add only the exposed (curved) surfaces; a bump or dent on a flat face changes area by +πr2+\pi r^2.
  • Volume of a combination: add the parts; subtract any cavity/depression.
  • Conversion (melt/recast/wire/water): volume is conserved — old volume == new volume; number of pieces =VbigVsmall=\dfrac{V_{\text{big}}}{V_{\text{small}}}; π\pi usually cancels.
  • Frustum (R,r,h)(R,r,h): l=h2+(R−r)2l=\sqrt{h^2+(R-r)^2}; CSA =πl(R+r)=\pi l(R+r); TSA =πl(R+r)+πR2+πr2=\pi l(R+r)+\pi R^2+\pi r^2; Volume =13πh(R2+r2+Rr)=\tfrac13\pi h(R^2+r^2+Rr).
  • 11 litre =1000=1000 cm3^3; 11 m3=106^3=10^6 cm3^3.

Common Traps to Avoid

  • Surface area vs volume rules: drop hidden faces for area, but add everything for volume.
  • Hemisphere TSA is 3πr23\pi r^2, not 2πr22\pi r^2 — include the flat circle when the base is exposed.
  • Cone: find ll first from l=r2+h2l=\sqrt{r^2+h^2} before using πrl\pi rl.
  • Combination heights: the part heights are not the total — e.g. cone height == total height −- hemisphere radius; cylinder length == capsule length −2r-2r.
  • Frustum slant uses (R−r)(R-r), not (R+r)(R+r).
  • Units: convert to one unit; capacities to litres (÷1000\div 1000 from cm3^3).

Final Tip: Sketch the solid, label each part's rr and hh, decide add or subtract, and keep π\pi symbolic until the end.