Area and Volume Units

Introduction

A chemist rarely measures a length for its own sake. Much more often, the question is "how much surface?" or "how much space?" How large is the surface of a catalyst? How much acid is in the burette? These questions call for units of area and volume . Both are derived units built from the metre, and understanding how they are built makes it easy to see why a litre, a cubic decimetre and a thousand cubic centimetres are all the same amount of liquid.

Core explanation

Area comes from length × length. The area of a rectangle is its length multiplied by its width. If both are measured in metres, the unit of area is metre × metre = m² (square metre). One square metre is the area of a square 1 m on each side. For smaller surfaces we use cm² or mm² . The small "2" reminds you that two lengths were multiplied together.

Volume comes from length × length × length. The volume of a box is length × width × height. With every length in metres, the unit of volume is m³ (cubic metre), the SI unit of volume. One cubic metre is the space inside a cube 1 m on each side — about the size of a large washing machine, and enough to hold 1000 kg of water. That is far too large for most laboratory work, so chemists use smaller cubes:

Unit Cube edge Everyday comparison --- --- --- m³ 1 m a large packing crate dm³ 10 cm (1 decimetre) a one-litre carton cm³ 1 cm a sugar cube mm³ 1 mm a grain of coarse salt

The litre. The litre (symbol L, sometimes l) is not an SI unit, but it is officially accepted for use alongside SI. It is defined as exactly 1 dm³ . A cube 10 cm on each side has a volume of 10 cm × 10 cm × 10 cm = 1000 cm³, so:

1 L = 1 dm³ = 1000 cm³ = 1000 mL

This means 1 mL is exactly 1 cm³ . Glassware in a school laboratory may be marked in either mL or cm³; the two labels mean the same volume. Concentrations in chemistry are usually quoted per dm³ (for example mol dm⁻³), which is the same as per litre.

Choosing a sensible unit. Pick a unit that gives a number that is easy to read. A drop from a dropping pipette is about 0.05 cm³; a burette delivers volumes of 0–50 cm³; a reagent bottle might hold 0.5 dm³; a laboratory gas cylinder might hold tens of dm³ of compressed gas; a swimming pool holds hundreds of m³.

Formulae

area of a rectangle = length × width (unit: m²)

volume of a cuboid = length × width × height (unit: m³)

1 L = 1 dm³ = 1000 cm³; 1 mL = 1 cm³

Step-by-step reasoning

To find the volume of a rectangular block in cm³:

1. Measure each of the three edges. 2. Make sure all three lengths are in the same unit — convert any mm or m into cm first. 3. Multiply the three lengths together. 4. Write the unit as cm³, because three centimetre lengths were multiplied. 5. If you need litres, divide the number of cm³ by 1000.

Visual explanation

Imagine a hollow 10 cm cube, the size of a one-litre carton. Now imagine filling it with sugar cubes 1 cm on each side. The bottom layer needs 10 rows of 10 cubes, which is 100 cubes, and there are 10 such layers. That makes 1000 small cubes filling one litre — the picture behind 1 dm³ = 1000 cm³.

Real-world analogy

Tiling a floor is like measuring area: you count how many square tiles cover the surface. Packing a removal van is like measuring volume: you count how many cube-shaped boxes fit inside. The tile and the box are the "unit" shapes, and the measurement simply says how many of them you need.

Real-world example

Car engine sizes are quoted in litres or cubic centimetres. A "1.6 litre" engine has a total cylinder volume of about 1600 cm³, and motorbike engines are commonly described as, for example, "125 cc", meaning 125 cubic centimetres. The same volume is being described with two different units.

Why?

Why is the litre defined as 1 dm³ rather than as some unrelated amount? Linking it directly to the metre means every volume unit fits one coherent system. A chemist can move between litres, cm³ and m³ just by multiplying or dividing by powers of ten, with no awkward conversion factors like those between pints and gallons.

Common misconception

"A millilitre measures liquids and a cubic centimetre measures solids." Both are units of volume and they are exactly equal. A 1 cm³ block of wood and 1 mL of water occupy precisely the same amount of space; the choice of name is only a matter of habit.

Worked example

Question: A rectangular glass tank is 40 cm long, 25 cm wide and 20 cm deep. What volume of water can it hold in cm³ and in litres?

Reasoning: volume = 40 cm × 25 cm × 20 cm = 20 000 cm³. Since 1 L = 1000 cm³, the volume in litres is 20 000 ÷ 1000 = 20 L.

Answer: 20 000 cm³, which is 20 L (or 20 dm³).

Quick check

1. How many cm³ are there in 0.25 dm³? Answer: 0.25 × 1000 = 250 cm³.

Exam focus

Remember the chain 1 dm³ = 1 L = 1000 cm³ = 1000 mL. Examiners often give volumes in cm³ but concentrations in mol dm⁻³, so you must divide cm³ by 1000 before calculating. Always include the squared or cubed index in the unit; "cm" alone is a length, not a volume.

Advanced insight

The litre has an unusual history. Between 1901 and 1964 it was defined as the volume of 1 kg of pure water at its maximum density, which turned out to be about 1.000028 dm³. The difference caused confusion in precise work, so in 1964 the litre was redefined as exactly 1 dm³. The symbol L (capital) is also allowed because a lower-case l is easily mistaken for the digit 1.

Summary

Area is length × length, measured in m², cm² or mm². Volume is length × length × length, with the SI unit m³ and the laboratory units dm³ and cm³. The litre is exactly 1 dm³, and 1 mL is exactly 1 cm³, so 1 L = 1000 cm³. Choose the unit that gives a convenient number, and always show the index in the unit.

Practice questions

1. What is the SI unit of volume, and what is the most common volume unit on laboratory glassware? Answer: The SI unit is the cubic metre (m³); glassware is usually marked in cm³ (or the equal unit mL). 2. A beaker contains 350 mL of water. Express this volume in cm³ and in dm³. Answer: 350 cm³, which is 350 ÷ 1000 = 0.350 dm³. 3. A rectangular sheet of metal is 12 cm long and 5.0 cm wide. What is its area, with units? Answer: 12 cm × 5.0 cm = 60 cm². 4. Explain why the unit of volume includes a small "3". Answer: Volume is found by multiplying three lengths together, so the length unit appears three times, for example cm × cm × cm = cm³.