Mensuration: Areas and Volumes
What the domain tests
Mensuration tests perimeter and area of plane figures, surface areas and volumes of solids, and combinations such as a cylinder with hemispherical ends.
Core rules and concepts
- 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.
- Cylinder: volume = πr²h; curved surface = 2πrh; total surface = 2πr(h + r).
- Cone: volume = ⅓πr²h; curved surface = πrl, where l = √(r² + h²).
- Sphere: volume = 4/3πr³; surface area = 4πr². Hemisphere: volume = 2/3πr³; curved surface = 2πr².
Worked example
A cylinder has radius 7 cm and height 10 cm. Its volume is πr²h = (22/7) × 49 × 10 = 1540 cm³. Its curved surface area is 2πrh = 2 × (22/7) × 7 × 10 = 440 cm². Its total surface area is 2πr(h + r) = 2 × (22/7) × 7 × 17 = 748 cm².
Traps that cost marks
Use the radius, not the diameter. Keep units consistent and write cm² for area and cm³ for volume. In a cone, slant height l is not the vertical height h. When solids are combined, subtract hidden surfaces only when the question asks for exposed area.