What is Area by counting? — Mathematics

Definition 28.5 Primary & Middle School Mathematics · Chapter 28 — Measures and Perimeter

The area of a figure drawn on a grid is the number of grid squares it covers. Two half-squares count as one. Area answers “how much surface”, perimeter answers “how long is the border” — two different questions!

Two figures on the grid: areas 8 and 6 squares. Now walk the borders and count the units: 12 for both! Same perimeter, different areas. — Two figures on the grid: areas $8$ and $6$ squares. Now walk the borders and count the units: $12$ for both! Same perimeter, different areas.

Examples

Example 28.6 (Half-squares)

A right triangle drawn on the grid with legs $4$ and $2$ covers $3$ full squares and $2$ half-squares … count carefully: its area is half of the $4 \times 2$ rectangle, i.e. $4$ squares. Cutting a rectangle along its diagonal always gives two triangles of equal area.

Example 28.7 (Same area, different perimeters)

A $4 \times 4$ square and an $8 \times 2$ rectangle both cover $16$ squares — same area. Borders: $16$ units for the square, $20$ for the rectangle. Neither measure determines the other: a long thin shape has lots of border for little surface.

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Examples

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