Lesson 63 of 100 | Quantitative Aptitude / Measurement / मापन
Length, area and volume unit conversions
Learning outcome
Convert length, area, volume and capacity units correctly, including calculations whose measurements initially use different units.
Concepts and assumptions
Length measures distance, area measures surface coverage, and volume measures occupied three-dimensional space. Their units must therefore scale differently.
Since 1 m = 100 cm, a square measuring 1 m on each side contains 100 × 100 = 10,000 squares of area 1 cm². Thus 1 m² = 10,000 cm², not 100 cm².
A cube with side 1 m contains 100 × 100 × 100 = 1,000,000 cubes of volume 1 cm³. Thus 1 m³ = 1,000,000 cm³.
Generally, if one larger length unit equals k smaller units, multiply by k for length, k² for area and k³ for volume. Reverse conversions require division. Identify the quantity before choosing the factor.
Useful exact identities are: 1 km = 1,000 m; 1 cm = 10 mm; 1 hectare = 10,000 m²; 1 L = 1,000 cm³; 1 mL = 1 cm³. Consequently, 1 m³ = 1,000 L. Capacity describes the usable internal volume of a container; litres do not measure mass.
Use one consistent unit before adding lengths or calculating areas and volumes. Conversion identities are exact, but converting a measured value does not improve its accuracy. Treat the exercise dimensions as ideal values; all answers here require no rounding.
Worked examples
Example 1 — Adding lengths. A route contains a 2.35 km section followed by a 480 m section. Find its total length in metres and kilometres.
First section = 2.35 × 1,000 = 2,350 m. Total = 2,350 + 480 = 2,830 m. Converting back: 2,830 ÷ 1,000 = 2.83 km.
Example 2 — Converting area. A rectangular mat is 1.8 m long and 75 cm wide. Find its area in m² and cm².
Width = 75/100 = 0.75 m. Area = 1.8 × 0.75 = 1.35 m². Area in cm² = 1.35 × 10,000 = 13,500 cm². Check using centimetres: length = 180 cm; 180 × 75 = 13,500 cm².
Example 3 — Container capacity. A rectangular tank has internal dimensions 80 cm × 50 cm × 45 cm. Find its full capacity in litres and m³.
Internal volume = 80 × 50 × 45 = 180,000 cm³. Capacity = 180,000/1,000 = 180 L. Volume = 180,000/1,000,000 = 0.18 m³. These are internal dimensions, so wall thickness is not subtracted.
Common mistakes
Applying a length factor to an area or volume; multiplying unlike length units without conversion; confusing m² with m³; treating litres as kilograms without density information.
Practice questions
- Express 0.72 km + 850 cm in metres.
- Convert 3.6 m² to cm² and 85,000 mm² to cm².
- Convert 0.045 m³ to litres and cm³.
- A 0.84-hectare plot is divided entirely into equal 350 m² plots, with no land reserved for paths. How many plots result?
Worked solutions
- 0.72 km = 720 m; 850 cm = 8.5 m. Total = 720 + 8.5 = 728.5 m.
- 3.6 × 10,000 = 36,000 cm². Since 1 cm² = 100 mm², 85,000/100 = 850 cm².
- Litres = 0.045 × 1,000 = 45 L. Cubic centimetres = 0.045 × 1,000,000 = 45,000 cm³.
- Total area = 0.84 × 10,000 = 8,400 m². Number of plots = 8,400/350 = 24; the units cancel because one area is divided by another.