Area and Perimeter - Relating Area and Boundary

Perimeter is the total length of the boundary of a closed figure. Imagine taking a piece of string and laying it exactly along the edges of a shape; the length of that string is the perimeter. It is measured in linear units like $cm$, $m$, or $km$.

Area represents the amount of surface or 'space' covered by a closed shape. Imagine a floor covered in identical square tiles; the number of tiles needed to cover the entire floor is its area. It is measured in square units such as $sq\ cm$ or $cm^{2}$.

Rectangles and squares are standard shapes where area and perimeter can be calculated using their dimensions. A rectangle has a length ($l$) and a width ($w$), while a square has four equal sides ($s$). Visually, the area of a rectangle is the number of unit squares that fit into its grid of $rows \times columns$.

Different shapes can have the same perimeter but different areas. For example, a rectangle with sides $6\ cm$ and $4\ cm$ has a perimeter of $20\ cm$ and an area of $24\ sq\ cm$. However, a very thin rectangle with sides $9\ cm$ and $1\ cm$ also has a perimeter of $20\ cm$, but its area is only $9\ sq\ cm$.

Conversely, shapes with the same area can have different perimeters. If you have 12 square tiles, you can arrange them in a $3 \times 4$ block (perimeter $14\ units$) or a long $1 \times 12$ line (perimeter $26\ units$). The more 'stretched out' a shape is, the larger its perimeter tends to be for the same area.

For irregular shapes drawn on a squared paper (grid), we estimate the area by counting the number of full squares and squares that are more than half-filled. We usually ignore squares that are less than half-filled. The perimeter is found by counting the number of unit segments along the outer edge of the shape.

The concept of 'Boundary' relates directly to fencing. If a problem asks how much wire is needed to go around a park, it is asking for the Perimeter. If it asks how much grass is needed to cover the park, it is asking for the Area.