Metric Units for Measurement of Area
Units of area are squares, so each conversion factor is a length factor squared: \(1\text{ m}^2 = 100 \times 100 = 10\,000\text{ cm}^2\). This NSW Year 8 maths subtopic (NESA outcome MA4‑ARE‑C‑01) links \(\text{mm}^2\), \(\text{cm}^2\), \(\text{m}^2\), hectares and \(\text{km}^2\), where a hectare is a square 100 m by 100 m.
You will convert in both directions, take two steps when there is no single factor, and put areas written in different units in order. You will also find the area of a phone screen in \(\text{cm}^2\), count the pavers needed for a courtyard, value a paddock at a price per hectare and work out the fertiliser for a rugby league field. It is part of Year 8 Area and Perimeter.
Theory
To change a unit of area, square the length factor: \(1\) m \(= 100\) cm, so \(1~\text{m}^2 = 100 \times 100 = 10\,000~\text{cm}^2\). A hectare is a square \(100\) m by \(100\) m, so \(1\) ha \(= 10\,000~\text{m}^2\). This Year 8 maths page shows how.
A unit of area is a square. Both its sides grow by the length factor, so the area factor is that factor squared.
\(1~\text{cm}^2\) is a square \(10\) mm by \(10\) mm, so \(1~\text{cm}^2 = 100~\text{mm}^2\).
\(1~\text{m}^2\) is a square \(100\) cm by \(100\) cm, so \(1~\text{m}^2 = 10\,000~\text{cm}^2\).
A hectare (ha) is a square \(100\) m by \(100\) m, so \(1\) ha \(= 10\,000~\text{m}^2\). \(1~\text{km}^2\) is a square \(1000\) m by \(1000\) m, so \(1~\text{km}^2 = 1\,000\,000~\text{m}^2 = 100\) ha.
Which way? Bigger unit to smaller unit: multiply (more small squares fit). Smaller unit to bigger unit: divide.
No single factor? Take two steps. From mm\(^2\) to m\(^2\), go mm\(^2\) \(\to\) cm\(^2\) \(\to\) m\(^2\).
Areas from lengths. Change the lengths to one unit and multiply, or multiply first and change the area unit; one checks the other. Comparing: change all the areas to one unit first.
Length factors and area factors:
| Length | Area |
|---|---|
| \(1\) cm \(= 10\) mm | \(1~\text{cm}^2 = 100~\text{mm}^2\) |
| \(1\) m \(= 100\) cm | \(1~\text{m}^2 = 10\,000~\text{cm}^2\) |
| \(100\) m by \(100\) m | \(1\) ha \(= 10\,000~\text{m}^2\) |
| \(1\) km \(= 1000\) m | \(1~\text{km}^2 = 1\,000\,000~\text{m}^2 = 100\) ha |
How to convert an area
- Find the area factor: square the length factor.
- Multiply going to a smaller unit; divide going to a bigger unit.
- Take two steps when there is no single factor, and check the size makes sense.
| \(420\,000 \div 10\,000\) | \(=\) | \(42\) |
\(420\,000~\text{m}^2 = 42\) ha
\(1~\text{m}^2 = 10\,000~\text{cm}^2\), so \(1~\text{cm}^2 = \dfrac{1}{10\,000}~\text{m}^2\).
| \(540 \div 10\,000\) | \(=\) | \(0.054\) |
\(540~\text{cm}^2 = 0.054~\text{m}^2\)
(i) \(3.6~\text{m}^2 = \underline{\qquad}~\text{cm}^2\)
(ii) \(82\,500~\text{m}^2 = \underline{\qquad}\) ha
| \(3.6 \times 10\,000\) | \(=\) | \(36\,000\) |
| \(82\,500 \div 10\,000\) | \(=\) | \(8.25\) |
(i) \(36\,000~\text{cm}^2\) (bigger to smaller unit: multiply)
(ii) \(8.25\) ha (smaller to bigger unit: divide)
| \(6\,300\,000 \div 100\) | \(=\) | \(63\,000~\text{cm}^2\) |
| \(63\,000 \div 10\,000\) | \(=\) | \(6.3\) |
\(6.3~\text{m}^2\)
| \(A\) | \(=\) | \(6.5 \times 3.4 = 22.1\) |
| \(22.1 \times 10\,000\) | \(=\) | \(221\,000\) |
\(22.1~\text{m}^2 = 221\,000~\text{cm}^2\). Check in centimetres: \(650 \times 340 = 221\,000\).
(i) Which reserve is larger?
(ii) By how many hectares, and how many square metres?
| \(0.52 \times 100\) | \(=\) | \(52~\text{ha}\) |
| \(52 - 38\) | \(=\) | \(14~\text{ha}\) |
| \(14 \times 10\,000\) | \(=\) | \(140\,000\) |
(i) Reserve A (\(52\) ha \(> 38\) ha)
(ii) \(14\) ha \(= 140\,000~\text{m}^2\)
Common pitfalls
Frequently asked questions
How many square centimetres are in a square metre?
\(10\,000\), because \(100 \times 100 = 10\,000\).
What is a hectare?
The area of a square \(100\) m by \(100\) m: \(10\,000~\text{m}^2\).
How many hectares are in a square kilometre?
\(100\).
How do I convert mm² to cm²?
Divide by \(100\), because \(1~\text{cm}^2 = 100~\text{mm}^2\).
Why do area conversions use squared numbers?
A unit of area is a square, and both of its sides change by the length factor.