Perimeter and area of rectangles

What is it about?

A rectangle has a length $a$ and a width $b$. Two questions sound similar but mean quite different things:

Area $A$ – how much surface does the rectangle cover? We measure it with unit squares. A square with side $1\,\text{cm}$ has an area of $1\,\text{cm}^2$ (one square centimetre). A rectangle with $a = 5\,\text{cm}$ and $b = 3\,\text{cm}$ holds $3$ rows of $5$ unit squares each, so $5 \cdot 3 = 15$ squares:

Perimeter $P$ – how long is the edge all the way round? The perimeter is a length and is given in $\text{cm}$. Walk along the four sides: $a + b + a + b$. Opposite sides are equally long, so

If you unroll the edge like a string, you get two equal pieces of length $a + b$ – that is where the bracket form $2 \cdot (a + b)$ comes from.

For a square, all sides are equally long: $A = a \cdot a$ and $P = 4 \cdot a$.

Same perimeter – different area. All rectangles with $P = 16\,\text{cm}$ have $a + b = 8\,\text{cm}$:

The more “square” the rectangle, the larger its area; the square has the largest. The other way round, with a fixed area long thin strips have the largest perimeter: for $A = 12\,\text{cm}^2$, $1\,\text{cm} \times 12\,\text{cm}$ has a perimeter of $26\,\text{cm}$, $3\,\text{cm} \times 4\,\text{cm}$ only $14\,\text{cm}$.

Composite shapes such as an L-shape can be split into rectangles ($A = A_1 + A_2$) or completed to a big rectangle, subtracting the missing piece ($A = a \cdot b - c \cdot d$). Surprisingly, the perimeter of the L-shape is exactly the same as that of the big rectangle.

Try it

1. Drag the corner of the rectangle. Watch how area and perimeter change. Which number grows faster when you make the rectangle bigger in both directions?
2. Press Tile and count: first the bottom row is laid square by square, then it is copied upwards. Use this to explain the formula $A = a \cdot b$. Rectangle 7 cm × 4 cm
3. Press Let the ant walk and count how many centimetres it walks. Afterwards the edge is unrolled as a string. Why are the two pieces of string equally long?
4. For the square 4 cm × 4 cm, $P = 16\,\text{cm}$ and $A = 16\,\text{cm}^2$. Why can you still not say that perimeter and area are “equal”?
5. Compare the long strip 10 cm × 1 cm with the square. Which has the larger perimeter, which the larger area?
6. Same perimeter: drag the corner along the dashed line. Which rectangle has the largest area? Go through all rectangles builds the whole bar chart.
7. Same area: which rectangle with $A = 24\,\text{cm}^2$ has the smallest perimeter? Also try A = 13 cm² – what is special about 13?
8. L-shape: find the area by splitting across, by splitting down and by completing to a rectangle. Then switch on Push the inner edges outwards. L-shape with pushed edges

Tasks

Task 1: Area and perimeter

A rectangle is $8\,\text{cm}$ long and $3\,\text{cm}$ wide. Work out its area and perimeter. Load task

Task 2: Working backwards

(a) A square has a perimeter of $20\,\text{cm}$. How long is one side, and what is its area? (b) A rectangle has an area of $18\,\text{cm}^2$ and is $6\,\text{cm}$ long. How wide is it, and what is its perimeter?

Task 3: True or false?

Lena says: “A rectangle with a larger perimeter also has a larger area.” Is she right?

Task 4: The largest flower bed

You have $24\,\text{m}$ of fencing for a rectangular flower bed. The sides should be whole metres. How long and how wide must the bed be so that it is as large as possible? (In the simulation, use $1\,\text{cm}$ for $1\,\text{m}$.) Load task

Task 5: L-shape in two ways

A $3\,\text{cm} \times 4\,\text{cm}$ piece is cut from the top right corner of an $8\,\text{cm} \times 6\,\text{cm}$ rectangle. Find the area in two different ways, and the perimeter. Load task

Ideas for the classroom

• Use: Grade 5, quantities and their units, section on area. Suitable for introducing the topic (tiling, the ant), for practice (task links with show=0&_hide=1) and for going further (comparisons, L-shape).
• Model: side lengths are whole centimetres ($a \le 10$, $b \le 6$). The squared paper shows whole centimetres in bold and half centimetres in fine lines – as in German exercise books, where one small square is $5\,\text{mm}$ wide. There, one square centimetre consists of four small squares. On screen, “1 cm” is of course not really 1 cm long.
• Representations: length $a$ (horizontal) and width $b$ (vertical) have the same colour in the picture, in the formulas and in the string. The string falls into two pieces of length $a + b$ and so justifies $P = 2 \cdot (a + b)$. In German, the perimeter is written $U$ (for Umfang).
• Common stumbling blocks:
  • “You can compare perimeter and area” – for the $4 \times 4$ square the numbers are equal, but the quantities (length and area) are not.
  • “A larger perimeter means a larger area” – see the Same perimeter mode.
  • Swapping the formulas ($P = a \cdot b$, $A = a + b$) or adding only two sides for the perimeter ($P = a + b$).
  • For the L-shape, the two inner edges are forgotten in the perimeter, or $A = a \cdot b$ is used.
  • On squared paper, every small square is counted as $1\,\text{cm}^2$.
• Outlook: converting units of area ($1\,\text{cm}^2 = 100\,\text{mm}^2$) and the area of parallelograms, triangles and trapeziums by splitting and completing in Year 6.