Area , counting unit squares

The area of a shape is the amount of flat space it covers. To make this precise, we choose a tiny square , say $1$ cm by $1$ cm , call it a unit square, and ask: how many such squares fit inside the shape?

Concept

Area is measured in square units because we are counting squares. The most common units:

• $1$ square centimetre ($1$ cm²) = the area of a square with side $1$ cm.
• $1$ square metre ($1$ m²) = the area of a square with side $1$ m.
• $1$ square kilometre ($1$ km²) = area of a square with side $1$ km.

Notice the small "²" , it is short for "square" and reminds us we are in two-dimensional territory. Writing "$30$ cm" when you mean "$30$ cm²" loses very important information.

For shapes that fit cleanly on a grid, area is easy: count the unit squares inside. A rectangle that covers $3$ rows of $5$ squares each has area $15$ square units. A square covering $4 \times 4 = 16$ unit squares has area $16$.

For shapes with curves or slants, some unit squares are only partly inside. There are two strategies:

1. Pair up partial squares. Half a square here and half a square there often combine into one whole. Useful for triangles and slanted shapes.
2. Estimate. Count the unit squares fully inside as $1$ each, those mostly inside as $1$, those at the edge as $\frac{1}{2}$, and ignore those barely touching. The result is a good approximation.

For familiar shapes there are formulas:

• Square (side $s$): area $= s^2$.
• Rectangle (length $l$, breadth $b$): area $= l \times b$.
• Triangle (base $b$, height $h$): area $= \frac{1}{2} b h$.

These formulas all come from counting unit squares; they are just shortcuts that work without drawing the grid.

Area vs perimeter

It is easy to mix these up. Here is a way to remember:

• Perimeter = walking around the edge. (Measured in length units.)
• Area = filling the inside. (Measured in square units.)

A long thin rectangle of $1$ cm by $20$ cm has perimeter $42$ cm but area $20$ cm². A square of $5$ cm by $5$ cm has perimeter $20$ cm but area $25$ cm². Different shapes, different stories.

Worked examples

Example 1. Find the area of a rectangle $7$ cm by $5$ cm.

• Area $= 7 \times 5 = 35$ cm².

Example 2. Find the area of a square with side $9$ m.

• Area $= 9^2 = 81$ m².

Example 3. A triangle has base $10$ cm and height $6$ cm. Find its area.

• Area $= \frac{1}{2} \times 10 \times 6 = 30$ cm².

Example 4. A garden of area $200$ m² is to be split into $2$-m by $2$-m square plots. How many plots fit?

• Each plot has area $4$ m².
• Number of plots $= 200 \div 4 = 50$.

Example 5. Estimate the area of a leaf placed on $1$-cm grid paper. The leaf covers about $12$ full squares and about $14$ partial squares.

• Full squares contribute $12$ cm².
• Partial squares contribute about $\frac{14}{2} = 7$ cm².
• Estimate: $12 + 7 \approx 19$ cm².

Example 6. Convert $5$ m² to cm².

• $1$ m $= 100$ cm, so $1$ m² $= 100 \times 100 = 10{,}000$ cm².
• $5$ m² $= 5 \times 10{,}000 = 50{,}000$ cm².

Try it yourself

1. Find the area of a rectangle $12$ cm by $8$ cm.
2. Find the area of a square with side $11$ m.
3. Find the area of a triangle with base $14$ cm and height $5$ cm.
4. A wall is $4$ m tall and $5$ m wide. How much paint is needed if $1$ litre covers $10$ m²?
5. Convert $3$ km² to m².
6. A rug measures $200$ cm by $150$ cm. Find its area in m².
7. A floor is to be tiled with $30$-cm-square tiles. The floor is $3$ m by $4$ m. How many tiles fit?
8. Compare: a rectangle of $1$ m by $9$ m and a square of side $3$ m. Which has the larger perimeter? Which has the larger area?

Activity

Foot-print area. Place your foot flat on a piece of grid paper (each square should be $1$ cm). Trace around it carefully. Count the full squares inside. Count the partial squares (count each as roughly $\frac{1}{2}$). Add to get the area of your footprint. Compare with your classmates , whose foot has the biggest area?