Metric System, Conversions, and Geometry
Questions here test unit conversion within the metric system and between metric and US customary units, temperature conversion, and perimeter, area, and volume of standard figures including circles and cylinders.
Metric prefixes as powers of ten
kilo 10³, hecto 10², deka 10¹, base unit 10⁰, deci 10⁻¹, centi 10⁻², milli 10⁻³, micro 10⁻⁶. Convert by multiplying by unit fractions so unwanted units cancel, or by moving the decimal the corresponding number of places.
Equivalences worth memorizing
1 in = 2.54 cm; 1 m ≈ 3.281 ft; 1 km ≈ 0.621 mi; 1 kg ≈ 2.205 lb; 1 L ≈ 1.057 qt; 1 gal ≈ 3.785 L. Temperature: °F = (9/5)°C + 32, °C = (5/9)(°F − 32), K = °C + 273.15.
Geometry formulas
Rectangle: A = lw, P = 2l + 2w. Triangle: A = ½bh, angles sum to 180°. Circle: C = 2πr = πd, A = πr². Trapezoid: A = ½(b₁ + b₂)h. Rectangular solid: V = lwh. Cylinder: V = πr²h, surface area 2πr² + 2πrh. Right triangles satisfy a² + b² = c².
Similar figures: if lengths scale by k, areas scale by k², and volumes by k³.
Worked example
A cylindrical can has radius 4 cm and height 10 cm.
- Volume: V = πr²h = π(4²)(10) = 160π ≈ 502.7 cm³.
- Since 1 cm³ = 1 mL, the volume is about 502.7 mL = 0.5027 L, roughly 0.50 L.
- Surface area: 2πr² + 2πrh = 2π(16) + 2π(4)(10) = 32π + 80π = 112π ≈ 351.9 cm².
Traps that cost marks
- Squared and cubed units: 1 m = 100 cm, but 1 m² = 10,000 cm² and 1 m³ = 1,000,000 cm³. Square or cube the conversion factor too.
- Temperature: the offset does not scale. A rise of 10 °C equals a rise of 18 °F, but 10 °C equals 50 °F.
- Radius versus diameter: halve the diameter before using πr² or πr²h.
- Rounding early: carry full precision and round only at the end.
- Similar figures: doubling the sides quadruples the area; do not apply the length factor to area.