To convert an area from one unit to another, square the length conversion factor. Since 1 m = 100 cm, then 1 m² = 100² = 10 000 cm². Forgetting to square it is the most common unit error in area questions.
This lesson belongs to area, perimeter and surface area. It connects with decomposing compound shapes, because multi-part figures can give some lengths in metres and others in centimetres.
Why does area convert differently from length?
A length is one-dimensional, so 1 m = 100 cm. An area is length × length, so both lengths change.
Picture a 1 m by 1 m square drawn on the floor. Measured in centimetres it is 100 cm by 100 cm, which is 10 000 small 1 cm squares. The factor 100 appears twice.
The conversions to know
| From | To | Multiply by |
|---|---|---|
| 1 m² | cm² | 10 000 |
| 1 cm² | mm² | 100 |
| 1 m² | mm² | 1 000 000 |
| 1 km² | m² | 1 000 000 |
To go the other way, divide by the same number.
Step by step
- Choose one unit for the whole question, usually the unit the answer needs.
- Convert every length before you calculate, not the area afterwards.
- Calculate the area with the usual formula.
- If you must convert an area, use the squared factor from the table.
- Write the unit with the answer, including the square.
Worked example
A rectangular floor is 4.5 m long and 320 cm wide. Square floor tiles measure 30 cm by 30 cm. How many tiles cover the floor, with none left over?
Step 1, one unit. Use metres for the floor: 320 cm = 3.2 m. The tile is 30 cm = 0.3 m.
Step 2, floor area. 4.5 × 3.2 = 14.4 m².
Step 3, tile area. 0.3 × 0.3 = 0.09 m².
Step 4, number of tiles. 14.4 ÷ 0.09 = 160 tiles.
Check in centimetres. Floor: 450 × 320 = 144 000 cm². Tile: 30 × 30 = 900 cm². 144 000 ÷ 900 = 160. Both routes agree.
The mistake to watch for
A student finds 14.4 m² and converts it to cm² by multiplying by 100.
Mistaken answer: 14.4 m² = 1440 cm²
This multiplies by the length factor 100 instead of the area factor 10 000.
The correct conversion is 14.4 × 10 000 = 144 000 cm². A quick sense check helps: a 1 m by 1 m square is 10 000 cm², so 14.4 of them must be far more than 1440.
Check yourself
1. Convert 2.5 m² to cm².
Show answer
2.5 × 10 000 = 25 000 cm².
2. Convert 450 000 mm² to cm².
Show answer
1 cm² = 100 mm², so divide by 100: 450 000 ÷ 100 = 4500 cm².
3. A rectangular plot is 0.6 km by 500 m. Find its area in m² and in km².
Show answer
0.6 km = 600 m. Area = 600 × 500 = 300 000 m². In km²: 300 000 ÷ 1 000 000 = 0.3 km². As a check, 0.6 × 0.5 = 0.3 km².
Where this leads next
Next, see how curved shapes work in distinguishing arc length from sector area. Use the mixed practice set to mix unit conversion with other shapes. The non-calculator working trainer is good for practising a division like 14.4 ÷ 0.09 by hand.
If you understand the squared factor but still slip under time pressure, a teacher can work through your own solutions and find the exact step. That is what we do in online one-to-one Mathematics tuition.