Using 9 unit squares, solve the following. 1. What is the smallest perimeter possible? 2. What is the largest perimeter possible? 3. Make a figure with a perimeter of 18 units. 4. Can you make other shaped figures for each of the above three perimeters, or is there only one shape with that perimeter? What is your reasoning?

Understand the Question A figure made of unit squares has an outer perimeter equal to the number of exposed side lengths. separate unit squares have a total of sides. When two squares share an edge, that edge is inside the figure, removing units from the outer boundary: Maximum shared edges give the smallest perimeter, while minimum shared edges give the largest perimeter. Part 1. What is the smallest perimeter possible? Step 1 · Calculate Smallest Perimeter To get the minimum perimeter, arrange the unit squares in the most compact shape, which is a square. In a square: Horizontal shared edges Vertical shared edges Total shared edges Answer > 1. Part 2. What is the largest perimeter possible? Step 1 · Calculate Largest Perimeter To get the maximum perimeter, minimize the number of shared edges by arranging the squares in a single continuous line of . In a line, there are shared edges between adjacent squares: Answer > 2. Part 3. Make a figure with a perimeter of 18 units. Step 1 · Determine Number of Shared Edges and Form Figure Using the perimeter formula: Any arrangement of squares with exactly shared edges (such as a C-shaped arrangement) gives a perimeter of . Answer > 3. A figure (such as a C-shape) with shared edges has a perimeter of . Part 4. Can you make other shaped figures for each of the above three perimeters, or is there only one shape with that perimeter? What is your reasoning? Step 1 · Analyze Uniqueness for Each Perimeter Perimeter of 12 units: There is only one shape (the square). A square is the most compact layout possible for 9 squares; any rearrangement reduces the number of shared edges () and increases the perimeter. Perimeter of 20 units: Multiple shapes are possible. Any tree-like arrangement of 9 squares where each new square joins by exactly 1 shared edge has shared edges and a perimeter of (e.g., a straight line, an L-shaped line, or a T-shaped strip). Perimeter of 18 units: Multiple shapes are possible. Different arrangements with exactly shared edges can be formed by rearranging squares while keeping the same count of shared sides. Answer > 4. There is only one shape ( square) for , while multiple shapes can be made for and . Common Mistakes Counting Shared Edges Once: Forgetting that each shared edge removes unit lengths from the total outer boundary (one from each touching square). Assuming 20 Units Requires a Straight Line: Assuming only a straight strip gives a perimeter of , whereas any arrangement with exactly shared edges (no closed loops) has the same perimeter.

Perimeter and Area | IT

Question 13

Using 9 unit squares, solve the following.

What is the smallest perimeter possible?
What is the largest perimeter possible?
Make a figure with a perimeter of 18 units.
Can you make other shaped figures for each of the above three perimeters, or is there only one shape with that perimeter? What is your reasoning?

Solution

Understand the Question

A figure made of unit squares has an outer perimeter equal to the number of exposed side lengths.
$9$ separate unit squares have a total of $9 \times 4 = 36$ sides.
When two squares share an edge, that edge is inside the figure, removing $2$ units from the outer boundary: $\text{Perimeter} = 36 - 2 \times (\text{number of shared edges})$
Maximum shared edges give the smallest perimeter, while minimum shared edges give the largest perimeter.

1. What is the smallest perimeter possible?

Step 1 · Calculate Smallest Perimeter

To get the minimum perimeter, arrange the $9$ unit squares in the most compact shape, which is a $3 \times 3$ square. Diagram 1

In a $3 \times 3$ square:

Horizontal shared edges $= 6$
Vertical shared edges $= 6$
Total shared edges $= 6 + 6 = 12$

\begin{aligned} P &= 36 - (2 \times 12) \\[0.6em] &= 36 - 24 \\[0.6em] &= 12 \text{ units} \end{aligned}

Answer

1. $12\text{ units}$

2. What is the largest perimeter possible?

Step 1 · Calculate Largest Perimeter

To get the maximum perimeter, minimize the number of shared edges by arranging the $9$ squares in a single continuous line of $1 \times 9$ . Diagram 2

In a $1 \times 9$ line, there are $8$ shared edges between adjacent squares:

\begin{aligned} P &= 36 - (2 \times 8) \\[0.6em] &= 36 - 16 \\[0.6em] &= 20 \text{ units} \end{aligned}

Answer

2. $20\text{ units}$

3. Make a figure with a perimeter of 18 units.

Step 1 · Determine Number of Shared Edges and Form Figure

Using the perimeter formula:

\begin{aligned} 18 &= 36 - (2 \times \text{shared edges}) \\[0.6em] 2 \times \text{shared edges} &= 36 - 18 = 18 \\[0.6em] \text{shared edges} &= 9 \end{aligned}

Any arrangement of $9$ squares with exactly $9$ shared edges (such as a C-shaped arrangement) gives a perimeter of $18\text{ units}$ . Diagram 3

Answer

3. A figure (such as a C-shape) with $9$ shared edges has a perimeter of $18\text{ units}$ .

4. Can you make other shaped figures for each of the above three perimeters, or is there only one shape with that perimeter? What is your reasoning?

Step 1 · Analyze Uniqueness for Each Perimeter

Perimeter of 12 units: There is only one shape (the $3 \times 3$ square). A square is the most compact layout possible for 9 squares; any rearrangement reduces the number of shared edges ( $<12$ ) and increases the perimeter.
Perimeter of 20 units: Multiple shapes are possible. Any tree-like arrangement of 9 squares where each new square joins by exactly 1 shared edge has $8$ shared edges and a perimeter of $20\text{ units}$ (e.g., a straight $1 \times 9$ line, an L-shaped line, or a T-shaped strip).
Perimeter of 18 units: Multiple shapes are possible. Different arrangements with exactly $9$ shared edges can be formed by rearranging squares while keeping the same count of shared sides.

Answer

4. There is only one shape ( $3 \times 3$ square) for $12\text{ units}$ , while multiple shapes can be made for $18\text{ units}$ and $20\text{ units}$ .

Common Mistakes

Counting Shared Edges Once: Forgetting that each shared edge removes $2$ unit lengths from the total outer boundary (one from each touching square).
Assuming 20 Units Requires a Straight Line: Assuming only a $1 \times 9$ straight strip gives a perimeter of $20\text{ units}$ , whereas any arrangement with exactly $8$ shared edges (no closed loops) has the same perimeter.

Question 13

More questions in IT

Split and rejoin

Area Maze Puzzles