Perimeter is not area

A common mistake: someone hears that two fields have the same fencing (same perimeter) and concludes they enclose the same amount of land. That isn't true. The relationship between perimeter and area is surprising, and once you see it you will never confuse the two again.

Concept

Two definitions, two ideas.

• Perimeter is the length of the boundary of a region. For a rectangle of length $L$ and width $W$, perimeter is $P = 2(L + W)$.
• Area is the count of unit squares that fit inside. For the same rectangle, area is $A = L \times W$.

They measure different things. Perimeter is a length (units like cm); area is a count of squares (units like $\mathrm{cm}^2$).

Same perimeter, different area. Consider rectangles with $P = 20$ cm. The condition $L + W = 10$ gives many options:

Same perimeter , but the area varies from $9 \mathrm{cm}^2$ all the way up to $25 \mathrm{cm}^2$! The square has the largest area among all rectangles of a fixed perimeter. The more lopsided the rectangle, the smaller its area.

Same area, different perimeter. Conversely, fix an area of, say, $36 \mathrm{cm}^2$:

Same area, perimeter swings from $74$ down to $24$. Again, the square is the most efficient , least perimeter for a given area.

Why this matters. Consider a farmer who wants to fence the largest possible field with a fixed length of fencing. The answer is a square (or, with no shape restriction, a circle, but among rectangles, the square). On the other hand, if you want to wrap a gift with the least possible wrapping paper for a fixed volume, you'd choose a cube-shaped box.

Quick checks.

• A long thin rectangle ($100$ cm by $1$ cm) has area $100 \mathrm{cm}^2$ but perimeter $202$ cm , huge boundary, modest area.
• A more square-like rectangle ($10$ cm by $10$ cm) has the same $100 \mathrm{cm}^2$ area but perimeter just $40$ cm.

So the same perimeter can enclose vastly different areas, and the same area can have wildly different perimeters. Perimeter and area answer different questions.

Worked examples

Example 1. Two rectangles: $\mathrm{R}_1$ is $8 \times 3$ and $\mathrm{R}_2$ is $7 \times 4$. Compare perimeters and areas.

• Perimeters: $2(8+3) = 22$ and $2(7+4) = 22$ , same.
• Areas: $24$ and $28$ , $\mathrm{R}_2$ has more area.

Example 2. A region has perimeter $24$ cm. What is its largest possible area as a rectangle? Smallest?

• Largest: square, side $6$ cm, area $36 \mathrm{cm}^2$. Smallest: very thin rectangle approaches $0$.

Example 3. Region $A$ is a $1 \mathrm{cm} \times 8 \mathrm{cm}$ rectangle; region $B$ is a square of side $4$ cm. Both have area $8$ vs $16$, but compare perimeters: $A$ has $18$ cm, $B$ has $16$ cm.

• The smaller rectangle has a larger perimeter than the square that contains twice the area.

Example 4. A wire $40$ cm long is bent into a rectangle. How long should the sides be to enclose the largest area?

• Each pair of sides totals $20$, so $L + W = 20$. Largest product when $L = W = 10$. Area $100 \mathrm{cm}^2$.

Try it yourself

1. Find two different rectangles with the same perimeter $24$ cm but different areas.
2. Two regions have the same area $36 \mathrm{cm}^2$. Sketch one with a small perimeter and one with a large one.
3. A rectangle has perimeter $30$ cm and length $10$ cm. Find its width and area.
4. Find the largest area enclosed by a $40$ cm wire bent into a rectangle.
5. Why does a $1 \mathrm{cm} \times 100 \mathrm{cm}$ rectangle have such a large perimeter?
6. Which has greater area: a $5 \times 4$ rectangle or a $6 \times 3$ rectangle?
7. Two rectangles, each with $P = 20$ cm, have areas $25$ and $9$. Sketch them.
8. True/false: "If one shape has a larger perimeter, it must have a larger area too." Explain.

Activity

Grid-paper experiment. On graph paper, draw $5$ different rectangles all with perimeter exactly $20$ cm (sides must be whole numbers). For each, write the length, width, and area. Now plot a small bar chart of these areas , the bar for the square ($5 \times 5$) should be the tallest. This is your first taste of optimisation: among many shapes of equal perimeter, the most regular one has the most area.