Rectangles and squares

How much rangoli powder do you need to colour a $7 \mathrm{cm} \times 4 \mathrm{cm}$ rectangle compared with an $8 \mathrm{cm} \times 3 \mathrm{cm}$ one? Both have the same perimeter of $22$ cm, yet they don't take the same amount of powder. That is because area, not perimeter, measures how much surface is enclosed.

Concept

The unit square. To measure area, we choose a square of side $1$ unit as our standard , a $1$ cm by $1$ cm square gives one square centimetre, written $\mathrm{cm}^2$. Area then means: how many of these unit squares fit inside the region?

Rectangles. Imagine a $7$ cm by $4$ cm rectangle. Cover it with rows of $1 \mathrm{cm}^2$ squares. Each row has $7$ squares, and there are $4$ rows. Total: $7 \times 4 = 28$ unit squares. So:

$\text{Area of a rectangle} = \text{length} \times \text{width}.$

This formula works for any length and width, including decimals or fractions. (A rectangle of $3.5$ cm by $2$ cm has area $7 \mathrm{cm}^2$ , half-squares pair up to fill in.)

Squares. A square is a special rectangle with all sides equal. If its side is $a$, then

$\text{Area of a square} = a \times a = a^2.$

Units of area. Always use squared units. Common ones:

• $\mathrm{cm}^2$ (square centimetres) , small regions like book covers.
• $\mathrm{m}^2$ (square metres) , rooms, gardens.
• $\mathrm{km}^2$ (square kilometres) , cities, countries.

Conversion: $1 \mathrm{m}^2 = 100 \mathrm{cm} \times 100 \mathrm{cm} = 10{,}000 \mathrm{cm}^2$. Notice it's $10{,}000$ , not $100$. When you scale lengths by $10$, area scales by $100$.

Doubling sides. If you double the side of a square, area becomes $(2a)^2 = 4a^2$ , four times bigger, not twice. Tripling the side gives $9$ times the area. This square law is one of the most important facts of area: area grows with the square of the side.

Rectangles can disguise themselves. Many shapes that aren't immediately rectangles can be cut into rectangles and put back together , like a flag made of three coloured stripes, each a rectangle of its own. The total area is just the sum of the parts.

A real-life check. When you buy floor tiles, count the area of the floor and the area of one tile. Number of tiles needed $=$ (floor area) / (tile area). The same idea decides how many seats fit on a stage, how much paint covers a wall, how many flowers fit in a bed.

Worked examples

Example 1. Find the area of a rectangle of length $12$ cm and width $9$ cm.

• Area $= 12 \times 9 = 108 \mathrm{cm}^2$.

Example 2. A square has side $7$ m. Find its area.

• Area $= 7^2 = 49 \mathrm{m}^2$.

Example 3. A floor measures $4$ m by $3$ m and is covered with square tiles of side $25$ cm. How many tiles?

• Floor area $= 4 \times 3 = 12 \mathrm{m}^2 = 120{,}000 \mathrm{cm}^2$.
• Tile area $= 25 \times 25 = 625 \mathrm{cm}^2$.
• Number $= 120000 / 625 = 192$ tiles.

Example 4. A rectangle's length is doubled while its width stays the same. By how much does the area change?

• Area was $L \times W$; new area $2L \times W = 2(LW)$. Area is doubled.

Try it yourself

1. Find the area of a $15$ cm by $8$ cm rectangle.
2. A square has side $11$ m. Area?
3. A wall is $4$ m high and $5$ m long. How many square metres of paint cover it?
4. A square photo frame has area $144 \mathrm{cm}^2$. Side length?
5. A rectangle has area $60 \mathrm{cm}^2$ and length $12$ cm. Width?
6. If you triple the side of a square, by what factor does the area grow?
7. How many $\mathrm{cm}^2$ make $1 \mathrm{m}^2$?
8. Two rectangles, one $8$ cm by $3$ cm and one $7$ cm by $4$ cm, have the same perimeter ($22$ cm). Which has more area, and by how much?

Activity

Tile your desk. Cut out $1$-cm squares from chart paper (about $30$ of them). Pick a small book and try to cover its top surface with them. Count the squares to estimate the book's surface area. Then measure the length and width of the book with a ruler and compute $L \times W$. How close are your two answers?