Question 13

The perimeter of a figure made of unit squares is the total length of its outer boundary. When squares are joined, their shared sides are no longer part of the perimeter. Each unit square has 4 sides.

Step 1 — Smallest Perimeter

To get the smallest perimeter, we arrange the 9 unit squares as compactly as possible. A square shape is the most compact. A $3 \times 3$ square uses 9 unit squares. If 9 squares were separate, they would have $9 \times 4 = 36$ sides. In a $3 \times 3$ square, some sides are shared between adjacent squares. There are 6 horizontal shared sides. There are 6 vertical shared sides. Total shared sides = $6+6 = 12$. Each shared side removes 2 unit lengths from the perimeter. So, the perimeter is $36 - (2 \times 12)$.

$P = 36 - (2 \times 12)$

$P = 36 - 24$

$\boxed{12 \text{ units}}$

Step 2 — Largest Perimeter

To get the largest perimeter, we arrange the 9 unit squares in a way that exposes as many sides as possible. This means arranging them in a long, thin strip. A $1 \times 9$ strip is the longest possible arrangement. If 9 squares were separate, they would have $9 \times 4 = 36$ sides. In a $1 \times 9$ strip, there are 8 shared sides between the 9 squares. Each shared side removes 2 unit lengths from the perimeter. So, the perimeter is $36 - (2 \times 8)$.

$P = 36 - (2 \times 8)$

$P = 36 - 16$

$\boxed{20 \text{ units}}$

Step 3 — Figure with Perimeter of 18 Units

We need to make a figure with 9 unit squares that has a perimeter of 18 units. If 9 squares were separate, they would have $9 \times 4 = 36$ sides. If the perimeter is 18 units, then $18 = 36 - (2 \times \text{shared sides})$. So, $2 \times \text{shared sides} = 36 - 18 = 18$. This means the figure must have 9 shared sides. The figure shown in the diagram (the C-shaped figure) has a perimeter of 18 units.

Step 4 — Other Shaped Figures

1. For 12 units perimeter: The $3 \times 3$ square is the only shape that gives the smallest perimeter of 12 units. Any other arrangement of 9 squares will expose more sides, leading to a larger perimeter. This is because a square is the most compact shape.

2. For 20 units perimeter: Other shaped figures can be made with a perimeter of 20 units. For example, a $1 \times 9$ strip of squares has a perimeter of 20 units. The 'C' shape shown in the diagram (which has 9 squares) also has a perimeter of 20 units. Many other arrangements can also achieve this perimeter.

3. For 18 units perimeter: Other shaped figures can be made with a perimeter of 18 units. The figure shown in Step 3 is one example. We can rearrange the squares to form different shapes while keeping 9 shared sides, which results in a perimeter of 18 units.

Answer

(i) The smallest perimeter is 12 units. (ii) The largest perimeter is 20 units. (iii) A figure with a perimeter of 18 units is shown in the diagram in Step 3. (iv) Other shaped figures for 20 and 18 unit perimeters can be made by rearranging the squares. There is only one shape (a $3 \times 3$ square) for the 12 unit perimeter.