Measurement and Spatial Figures
What this domain tests
Area and perimeter of plane figures and surface area and volume of prisms, cylinders, cones and spheres, including the effect of enlarging a solid.
Core rules
- Area: triangle ½bh; trapezium ½(a+b)h; circle πr², circumference 2πr.
- Prism or cylinder: V = (base area) × h; cylinder surface = 2πr² + 2πrh.
- Cone: V = ⅓πr²h, slant height l = √(r²+h²), lateral surface πrl, total surface πr(l+r).
- Sphere: V = 4/3 πr³, surface 4πr².
- Similar solids with length ratio k have surface areas in ratio k² and volumes in ratio k³.
Worked example
A cone has radius 3 cm and height 4 cm. Slant height l = √(9+16) = 5 cm. Volume = ⅓ × π × 9 × 4 = 12π cm³. Total surface = πrl + πr² = 15π + 9π = 24π cm².
Traps that cost marks
Using the perpendicular height instead of the slant height in πrl; omitting the circular base from a total-surface answer; assuming doubling the radius doubles the volume (it multiplies it by 8); mixing units such as cm and m inside one formula.