Area of a Rectangle
Area is covering with squares
Area measures how much surface a flat shape covers, and it is counted in unit squares. If a rectangle can be tiled exactly by squares one centimetre on each side, its area is the number of those squares that fit inside. A rectangle four squares wide and three high holds twelve squares, so its area is twelve square centimetres. Thinking of area as covering, as filling a shape with squares that leave no gaps and do not overlap, is the picture that makes every area formula sensible rather than something to memorise.
Rows times columns
Counting the squares one at a time is slow, but a rectangle arranges them into neat rows and columns. A rectangle five wide and three high has three rows of five squares, and three fives is fifteen. There is no need to count each square: the number of columns multiplied by the number of rows gives the total at once. This is the heart of the area formula. The two side lengths are exactly the number of squares along each direction, so multiplying them counts the whole array.
The formula for a rectangle
Because the squares line up in rows and columns, the area of any rectangle is its width multiplied by its height. This is the formula: area equals width times height. A rectangle eight centimetres by five centimetres has an area of eight times five, which is forty square centimetres, found in a single step with no tiling required. The formula works for any rectangle whatever its size, turning a measurement of two lengths into a measurement of the surface between them, and it is the first area formula every other one is built upon.
Square units
Area is always measured in square units, because it counts squares. A square centimetre is the area of a square one centimetre on each side, and a square metre is a square one metre on each side. The right unit depends on what is being measured: a postage stamp is a few square centimetres, a classroom floor a few square metres, a paddock measured in hectares. Choosing a sensible square unit keeps the numbers manageable, and writing area with the little raised two, as in square centimetres, signals that two lengths have been multiplied.
Composite shapes
Many shapes are not single rectangles but can be split into them. An L-shape divides into two rectangles; finding the area of each and adding them gives the area of the whole. This is how the rectangle formula reaches beyond plain rectangles: any shape made of rectangular pieces can be measured by breaking it apart, applying the formula to each piece, and summing. The skill of splitting a composite shape sensibly, then adding the parts, is what makes the single formula for a rectangle so widely useful in real problems.
Working backwards
The area formula can also be run in reverse. If the area of a rectangle is known along with one side, the other side is found by dividing: a rectangle of area thirty-five with one side five has an other side of thirty-five divided by five, which is seven. This is the inverse of multiplying the sides, and it solves a different kind of practical problem, such as working out how long a garden bed must be to cover a set area. Reading the formula both forwards and backwards doubles the range of questions it can answer.
Where area leads
The area of a rectangle is the foundation of all area work. From it come the areas of triangles and parallelograms, which are found by relating each shape back to a rectangle, and the surface areas of solids, which add up the rectangular faces. The idea that area counts unit squares, and that for a rectangle this is simply width times height, carries forward into every later measurement topic, from the area of a circle to the volumes that build on area as their base.