Derived Quantities: Area, Volume, and Unit Conversions
A derived quantity is a quantity obtained by combining base quantities, such as area (length × length, in m2) and volume (length × length × length, in m3); a unit conversion changes the same quantity from one unit to another by multiplying by a conversion factor equal to 1. In brief: to convert units, multiply by a fraction that equals 1, such as (1 m ÷ 100 cm), and cancel the unit you do not want. Areas and volumes convert by the square and cube of the length factor: 1 m2 = 10,000 cm2 and 1 m3 = 1,000,000 cm3.
Most quantities used in physics are combinations of length, mass and time. Being able to calculate areas and volumes and to change units correctly is a basic skill for every problem. This guide explains distance, area and volume, shows the method of unit conversion, gives a table of common conversions, describes estimation, works through examples and corrects common mistakes.
What are distance, area and volume?
- Distance (length): how far apart two points are, measured along a path. The SI unit is the meter. The distance traveled along a curved path is longer than the straight-line separation.
- Area: the size of a surface. For a rectangle, A = length × width. For a triangle, A = ½ × base × height. For a circle, A = πr2. The SI unit is m2.
- Volume: the amount of space that a body occupies. For a box, V = length × width × height. For a cylinder, V = πr2h. For a sphere, V = (4/3)πr3. The SI unit is m3, but the liter (1 L = 1,000 cm3 = 10−3 m3) and the milliliter (1 mL = 1 cm3) are used for liquids.
- Irregular shapes: area can be found by counting squares on a grid, and volume by the displacement of water.
How do you convert units?
Use a conversion factor, a fraction in which the numerator and denominator are equal amounts in different units, such as 1 m / 100 cm = 1. Multiplying by it does not change the quantity, only its unit.
- Write the starting quantity with its unit.
- Choose a factor in which the unit to remove is in the denominator.
- Multiply and cancel the units.
- Check that the size of the number is sensible: converting to a smaller unit makes the number larger.
For compound units, convert each part. For example, 90 km/h = 90 × (1,000 m ÷ 3,600 s) = 25 m/s. For areas and volumes, the factor is squared or cubed: since 1 m = 100 cm, 1 m2 = (100 cm)2 = 10,000 cm2, and 1 m3 = (100 cm)3 = 106 cm3. Consequently, a density of 1 g/cm3 equals 1,000 kg/m3.
Common conversions
| Quantity | Conversion |
|---|---|
| Length | 1 km = 1,000 m; 1 m = 100 cm = 1,000 mm |
| Area | 1 m2 = 104 cm2; 1 hectare = 10,000 m2 |
| Volume | 1 m3 = 106 cm3 = 1,000 L; 1 L = 1,000 mL = 1,000 cm3 |
| Time | 1 h = 3,600 s; 1 day = 86,400 s |
| Speed | 1 m/s = 3.6 km/h |
| Mass | 1 kg = 1,000 g; 1 tonne = 1,000 kg |
| Density | 1 g/cm3 = 1,000 kg/m3 |
How do you estimate measurements?
An estimate is a rough value found by simple reasoning. It checks whether a calculated answer is sensible, and it helps when exact data are not available. To estimate, compare with something familiar and round the numbers. A door is about 2 m high, a step is about 0.7 m long, an apple has a mass of about 0.1 kg, and a 1 L bottle of water has a mass of about 1 kg. For a quantity like the number of grains of sand in a bucket, estimate the volume of one grain and divide it into the volume of the bucket. Estimates are usually given to one significant figure, or as an order of magnitude (a power of ten).
Worked example: area, volume and conversions
- Area of a room: a floor is 5.0 m by 3.0 m. A = 5.0 × 3.0 = 15 m2. In cm2: 15 × 10,000 = 150,000 cm2 = 1.5 × 105 cm2.
- Volume of a box: a box is 20 cm × 10 cm × 5 cm. V = 20 × 10 × 5 = 1,000 cm3. In liters: 1,000 cm3 ÷ 1,000 = 1.0 L. In m3: 1,000 ÷ 106 = 1.0 × 10−3 m3.
- Speed: convert 90 km/h to m/s: 90 × 1,000 m ÷ 3,600 s = 25 m/s. Conversely, 10 m/s × 3.6 = 36 km/h.
- Density: a metal has a density of 2.5 g/cm3. In kg/m3: 2.5 × 1,000 = 2,500 kg/m3. Check by steps: 2.5 g = 2.5 × 10−3 kg and 1 cm3 = 10−6 m3, so the density is 2.5 × 10−3 ÷ 10−6 = 2.5 × 103 kg/m3.
- Circle: a circular table has a radius of 0.60 m. A = π × 0.602 = π × 0.36 = 1.13 m2.
How did units develop?
Early units were based on the body, such as the cubit (elbow to fingertip) and the foot, and varied from place to place. The French Revolution led to the metric system in the 1790s, with the liter defined as the volume of a cube with 10 cm sides. The hectare, from the Latin area, was introduced in the same system. Imperial and US customary units, such as the inch (defined as exactly 2.54 cm since 1959), the gallon and the mile, are still used in some countries.
Which mistakes are common?
Mistake 1: Converting areas with the linear factor
1 m2 is 10,000 cm2, not 100 cm2. The factor must be squared.
Mistake 2: Multiplying instead of dividing
Check the direction: a smaller unit gives a larger number.
Mistake 3: Mixing units in a formula
Do not multiply meters by centimeters. Convert first.
Mistake 4: Confusing volume units with capacity units
1 mL equals 1 cm3, and 1 L equals 1,000 cm3 (not 100 cm3).