Concepts: Area (part 1)

1.1 Why area is counted in unit squares

Rangoli art fills regions of different shapes with coloured powder. If the powder is spread evenly, the region that needs more powder is the region with the larger area. To compare regions, we choose a small square as the unit and count how many copies of it fit inside a region without gaps or overlaps. A square of side 1 cm is a convenient unit for paper-sized regions, and its area is one square centimetre, written 1 sq cm or 1 cm².

For a rectangle with length 7 cm and width 4 cm, seven columns of four unit squares fit, giving 7 × 4 = 28 unit squares, so the area is 28 cm². A rectangle of length 8 cm and width 3 cm holds 24 unit squares, so its area is 24 cm². The first rectangle needs more rangoli powder.

The choice of unit matters only for the label; the comparison works with any square unit. Paper is measured in square centimetres, rooms in square metres, and land in square feet or acres. A larger unit simply packs many small squares into one, so the same region gets a smaller number when the unit is bigger. This is why a careful answer always names the unit of area. A count that ends in half a square is still a valid area; the region simply contains a fraction of a unit square, and the fraction can be measured by cutting the square into smaller pieces.

Counting squares also works when the region is not a rectangle. For a triangle, a parallelogram or a trapezium, the counting is awkward, so we cut the shape into pieces that are easy to count and then add or subtract. The formulas in this chapter are short because each one records a clever way of rearranging a shape into a rectangle. Every formula should be read together with that picture, because the picture explains why the formula has a half, a product or a sum of sides.

The pattern of the count gives the rule: area of a rectangle = length × width. A square is a rectangle whose sides are equal, so its area is side × side. Fractions of a unit square are allowed as well; a unit square can be cut into halves or quarters, which is why the answer for a triangle can be a half-number.

1.2 The diagonal cuts a rectangle in half

Draw a diagonal of a rectangle. It divides the rectangle into two congruent right triangles, so each of the two triangles carries half of the rectangle's area. For the 7 cm by 4 cm rectangle, each triangle has area (7 × 4) ÷ 2 = 14 cm².

The same picture works for any right triangle: it is half of a rectangle whose length and width are its base and its height. That gives the formula area of a triangle = half the base times the height. The height must be measured perpendicular to the base, because the formula counts how many unit squares fit in the rectangle and then takes half.

1.3 Perimeter is not a measure of area

The perimeter of a region is the length of its boundary. Two regions with the same perimeter can have very different areas, so perimeter cannot replace area. A 6 cm by 4 cm rectangle has perimeter 20 cm and area 24 cm², while an 8 cm by 3 cm rectangle has perimeter 22 cm and the same area 24 cm². The second shape has the longer boundary but no extra area.

The opposite can also happen: a 20 cm by 1 cm strip has perimeter 42 cm and area only 20 cm², while a 6 cm by 6 cm square has perimeter 24 cm and area 36 cm². The strip has the longer perimeter but the smaller area. A long thin shape spends its area along its boundary instead of filling the inside. This is why area is measured by counting unit squares, not by measuring the boundary. A square 5 cm by 5 cm has area 25 cm² and perimeter 20 cm, while a rectangle 25 cm by 1 cm has the same area but a perimeter of 52 cm; the long rectangle has much more boundary for the same amount of inside. The reverse also happens, so neither measure can stand in for the other. A quick way to estimate an irregular area is to draw it on grid paper, count the complete squares, add half of the squares that are crossed by the boundary, and multiply by the area of one grid square. The estimate is not exact, but it shows how area grows with the number of squares rather than with the length of the outline.