Metric Units for Measurement of Area
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.