Mensuration of Plane Figures and Solids
Mensuration asks for perimeter, area, surface area and volume, often of composite figures. The formulas are supplied in some papers, but choosing the correct one and keeping units consistent is the real test.
Core rules
- Triangle area = ½ × base × height; Heron's formula uses s = (a + b + c)/2 and area = √[s(s − a)(s − b)(s − c)].
- Circle: area = πr², circumference = 2πr. Sector area = (θ/360)πr²; arc length = (θ/360)2πr.
- Cuboid: volume = lbh, total surface area = 2(lb + bh + hl). Cube: volume = a³, surface area = 6a².
- Cylinder: volume = πr²h, curved surface = 2πrh, total surface = 2πr(r + h).
- Cone: volume = ⅓πr²h, curved surface = πrl, where l² = r² + h². Sphere: volume = (4/3)πr³, surface = 4πr².
Worked example
A cylinder has radius 7 cm and height 10 cm. Volume = πr²h = (22/7) × 49 × 10 = 1540 cm³. Curved surface area = 2πrh = 2 × (22/7) × 7 × 10 = 440 cm². Total surface area = 2πr(r + h) = 44 × 17 = 748 cm².
Traps
Diameter is not radius: halve it before substituting. For a cone, l is the slant height, not the vertical height. Convert all measurements to the same unit before calculating. When a solid is melted and recast, volume is conserved but surface area is not.