Mensuration: Areas and Volumes
This section covers Mensuration: Areas and Volumes. The exam tests surface areas and volumes of cylinders, cones, spheres, hemispheres, and frustums, including combinations of solids.
Core formulas
- Cylinder: curved surface area = 2πrh; total surface area = 2πr(r+h); volume = πr²h.
- Cone: curved surface area = πrl; total surface area = πr(l+r); volume = ⅓πr²h, where l² = r²+h².
- Sphere: surface area = 4πr²; volume = 4/3πr³. Hemisphere: curved area = 2πr²; total area = 3πr²; volume = 2/3πr³.
- Frustum: volume = ⅓πh(r₁² + r₁r₂ + r₂²); slant height l = √[h² + (r₁ - r₂)²]. When solids are joined, add volumes but subtract hidden surfaces from total surface area.
Worked example
Find the volume of a cylinder with radius 7 cm and height 10 cm, using π = 22/7. Volume = πr²h = (22/7)×7²×10 = (22/7)×49×10 = 22×7×10 = 1540 cm³.
Traps
Keep units consistent; convert diameters to radii before substituting. Curved surface area and total surface area are different. For a cone, l is not h. When a hemisphere is mounted on a cylinder, the circular joint is not part of the exposed surface. Do not use r instead of r² in volume formulas.