Mensuration
Mensuration tests area and perimeter of plane figures and surface area and volume of solids: cuboid, cube, prism, cylinder, cone, sphere, hemisphere, and pyramid. SEE questions often combine a formula with unit conversion or ask for the cost of painting or plastering a surface.
Core rules
- Triangle area = ½×base×height; Heron’s formula A=√[s(s−a)(s−b)(s−c)], s=(a+b+c)/2.
- Parallelogram = base×height; trapezium = ½(sum of parallel sides)×height; rhombus = ½d₁d₂; circle = πr², circumference 2πr.
- Cuboid: volume lbh, surface area 2(lb+bh+hl). Cube: volume a³, surface area 6a².
- Cylinder: volume πr²h; curved surface 2πrh; total surface 2πr(r+h).
- Cone: slant height l=√(r²+h²); volume ⅓πr²h; curved surface πrl; total πr(r+l).
- Sphere: volume 4/3πr³, surface 4πr². Hemisphere: volume 2/3πr³, total surface 3πr².
- Prism/pyramid: prism volume = base area×height; pyramid volume = ⅓×base area×height.
Worked example
A closed cylinder has r=7 cm and h=10 cm. Volume = πr²h = (22/7)×49×10 = 1,540 cm³. Curved surface = 2πrh = 2×(22/7)×7×10 = 440 cm². Total surface = 2πr(r+h)=2×(22/7)×7×17=748 cm². Using r=7 makes 22/7 convenient; otherwise keep π or use 3.14 consistently.
Traps
- Confusing height h with slant height l in a cone.
- Using 2πrh for total cylinder area and omitting the two circular ends.
- Forgetting that a hemisphere’s total surface includes the flat circular base: 3πr², not 2πr².
- Mixing cm and m without converting.
- Using diameter in place of radius.