A square meter
An area counts unit squares. A square meter is a square 1 m long on each side, and it is written . How many square centimeters fit inside it?
Each side of the square is 1 m, which is 100 cm. So along the top edge there is room for a row of 100 centimeter squares, and down the side there is room for 100 of those rows. That is 100 × 100 = 10,000 squares, so a square meter is 10,000 cm².
It is not 100 cm². A row of 100 squares covers only the thin strip along one edge. The square is 100 rows deep.
1 m × 1 m = 1 m²: one tile of the whole square
Cut the meter square into centimeters and count the tiles
The square is 1 m on each side. Cut it into centimeters, and each side is cut into 100, so the square holds 100 × 100 = 10,000 small squares. The lens shows one corner close up.
Once for each direction
A length has one direction, so it takes the factor once: 1 m = 100 cm. An area has two directions, the length and the width, and each of them is changed to centimeters. So an area takes the factor twice: a square meter is 100 × 100 = 10,000 cm².
Then 3 square meters are 3 × 10,000 = 30,000 cm². Going back, divide by 10,000: 50,000 cm² is 50,000 ÷ 10,000 = 5 square meters.
Square centimeters and square millimeters
The same square works for any pair of units. A centimeter is 10 mm, so a square centimeter is a square 10 mm long on each side. It holds 10 rows of 10 square millimeters: 1 cm² = 10 × 10 = 100 mm².
A square centimeter, cut into 10 rows of 10 square millimeters: 100 of them.
Change the lengths first
A floor is 4 m long and 3 m wide, so its area is 4 × 3 = 12 square meters. To find it in square centimeters, there are two ways, and they agree.
Change the area: 12 square meters are 12 × 10,000 = 120,000 cm². Or change the two lengths first and then multiply: 4 m = 400 cm and 3 m = 300 cm, so the area is 400 × 300 = 120,000 cm². Changing the lengths first is the safer way, because each length takes the factor 100 once and the multiplying does the rest.
The floor is 3 rows of 4 square meters, 12 square meters in all. Each of those squares holds 10,000 cm².
Volume takes it three times
A volume has three directions: length, width and height. A cubic meter, written , is a cube 1 m on every edge, and every edge is 100 cm. So the factor is used three times: 100 × 100 × 100 = 1,000,000, and a cubic meter is 1,000,000 cm³.
So a length takes the factor once, an area takes it twice and a volume takes it three times. Taking it only once, and writing a square meter as 100 cm², is the most common mistake.
Every edge of a cubic meter is 100 cm, so it holds 100 × 100 × 100 = 1,000,000 cubic centimeters.
Worked example: Square Tiles on a Floor Measured in Meters
Question A rectangular floor is 4 m long and 3 m wide. It is to be covered completely with square tiles of side 25 cm. (a) What is the area of the floor in square centimeters? (b) How many tiles are needed?
1.Change the length and the width of the floor to centimeters: 4 m = 400 cm and 3 m = 300 cm.
The tile is measured in centimeters, so the floor is too: 400 cm by 300 cm. 2.(a) The area of the floor is 400 × 300 = 120 000 cm².
(a) The area of the floor is 400 × 300 = 120 000 cm². 3.The area of one tile is 25 × 25 = 625 cm².
The gold square is one tile. Its area is 25 × 25 = 625 cm². 4.(b) The number of tiles needed is 120 000 ÷ 625 = 192. Check: 400 ÷ 25 = 16 tiles fit along the length and 300 ÷ 25 = 12 along the width, and 16 × 12 = 192.
(b) 120 000 ÷ 625 = 192 tiles. There are 16 along the length and 12 along the width.
Answer: (a) 120 000 cm²; (b) 192 tiles
Common mistakes
- Changing 12 m² to 1200 cm² by multiplying by 100. A square meter is 100 cm long and 100 cm wide, so each square meter is 100 × 100 = 10 000 cm².
- Dividing the area of the floor by 25, the side of a tile. Each tile covers an area of 25 × 25 = 625 cm², so the area of the floor must be divided by 625.