Surface Areas and Volumes
Surface Areas and Volumes tests formulas for solids, combinations of solids, and unit conversions. Questions often ask for total surface area, curved surface area, volume, or the cost of painting or filling.
Core rules
- Cuboid: volume lbh; total surface area 2(lb+bh+hl). Cube: volume a^3; total surface area 6a^2.
- Cylinder: curved surface 2πrh; total surface 2πr(r+h); volume πr^2h.
- Cone: slant l = sqrt(r^2+h^2); curved surface πrl; total surface πr(l+r); volume (1/3)πr^2h.
- Sphere: surface 4πr^2; volume (4/3)πr^3. Hemisphere: curved surface 2πr^2; total surface 3πr^2; volume (2/3)πr^3.
- For combined solids, add exposed surfaces only; volumes of joined parts add.
Worked example
A cylinder has radius 7 cm and height 10 cm. Its volume is πr^2h = (22/7)×49×10 = 1540 cm^3. Its curved surface area is 2πrh = 2×(22/7)×7×10 = 440 cm^2. Total surface area is 2πr(r+h)=2×(22/7)×7×17=748 cm^2, including both circular ends.
Traps
Do not use diameter in place of radius. Curved surface area excludes bases; total surface area includes them. For a cone, l is not h. When solids are joined, hidden faces are not part of the exposed surface area. Convert units before calculating volume or cost.