Rectangles, squares, and why perimeter is not area

How many ways can you divide a square into four parts of equal area? Four strips, four small squares, four triangles meeting at the centre, and then infinitely many more. Take any division, push one part's edge inwards and another part's edge outwards by the same amount: what one part loses, the other gains, and all four areas stay equal.

Now a practical question. In rangoli, shapes are filled evenly with coloured powder. Which rectangle needs more powder: 7 cm by 4 cm, or 8 cm by 3 cm?

Answer it by counting how many unit squares (squares of side 1 cm, not overlapping) fit inside each. The first holds 7 × 4 = 28, the second 8 × 3 = 24. So the 7 by 4 rectangle needs more powder, even though the 8 by 3 one looks longer. Count, don't guess.

That is what area is: the number of unit squares (parts of squares are allowed) that exactly cover the shape. In a rectangle they line up in rows and columns, which is why area of a rectangle = length × width.

We write the answers as 28 sq. cm and 24 sq. cm, or as 28 cm² and 24 cm².

A diagonal cuts a rectangle into two congruent triangles (exact copies), so each takes half the unit squares. For the 7 by 4 rectangle, each triangle is ½ × 7 × 4 = 14 cm².

Why all this counting? Couldn't we just use the perimeter (the length of the boundary) to measure how much space is inside?

No. Two shapes can have the same perimeter and very different areas. You can even find a shape with the larger perimeter and the smaller area.

The boundary and the inside are simply different measurements.