Perimeter and Area | IT

Question 15

Experiment placing this new square at different places and think what the change in perimeter will be. Can you place the square so that the perimeter: a) increases; b) decreases; c) stays the same?

Solution

Understand the Question

The original plus-shaped figure is formed by 5 unit squares, each with side length of $1\text{ unit}$ .
The perimeter is the total length of the outer boundary: $\text{Original perimeter} = 4 \times 3\text{ units} = 12\text{ units}$
When adding a new square that touches $k$ $k$ sides of the existing shape:
- It adds $(4 - k)$ outer sides.
- It covers $k$ existing outer sides.
- $\text{Change in perimeter} = (4 - k) - k = 4 - 2k\text{ units}$ .

a) Can you place the square so that the perimeter increases?

Step 1 · Find Original Perimeter

Let each side of a small square be $1\text{ unit}$ . The plus sign consists of $5$ squares where each of the $4$ outer squares has $3$ exposed sides. Diagram 1

\begin{aligned} \text{Total perimeter} &= 3 \text{ units} + 3 \text{ units} + 3 \text{ units} + 3 \text{ units} \\[0.6em] &= 4 \times 3 \text{ units} \\[0.6em] &= 12 \text{ units} \end{aligned}

Step 2 · Place Square Touching 1 Side

To increase the perimeter, place the new square touching only $1\text{ side}$ (e.g., above the top square). This adds $3$ new sides and covers $1$ existing side, giving a net increase of $2\text{ units}$ . Diagram 2

\begin{aligned} \text{New perimeter} &= 12 \text{ units} + 2 \text{ units} \\[0.6em] &= 14 \text{ units} \end{aligned}

Answer

a) Yes, the perimeter increases to $14\text{ units}$ (an increase of $2\text{ units}$ ) when the square touches $1\text{ side}$ .

b) Can you place the square so that the perimeter decreases?

Step 1 · Check Possibility to Decrease Perimeter

To decrease the perimeter, the new square would need to touch $3\text{ sides}$ (adding $1$ side while removing $3$ sides):

$\text{Change} = 1 - 3 = -2 \text{ units}$ Diagram 3

Since the plus sign has only outer corners and no internal pocket surrounded on $3$ sides, a new square cannot touch $3$ sides. Therefore, it is impossible to decrease the perimeter.

Answer

b) No, it is not possible to decrease the perimeter for this shape.

c) Can you place the square so that the perimeter stays the same?

Step 1 · Place Square Touching 2 Sides

To keep the perimeter the same, place the new square in a corner where it touches $2\text{ sides}$ . This adds $2$ new outer sides and covers $2$ existing sides, resulting in a net change of $0\text{ units}$ . Diagram 4

\begin{aligned} \text{New perimeter} &= 12 \text{ units} + 0 \text{ units} \\[0.6em] &= 12 \text{ units} \end{aligned}

Answer

c) Yes, the perimeter stays the same at $12\text{ units}$ when the square is placed in an inner corner touching $2\text{ sides}$ .

Common Mistakes

Assuming Added Area Always Increases Perimeter: Adding a square adds area, but perimeter depends solely on the boundary. If $2$ sides are glued, the net change in perimeter is zero.
Counting Glued Sides Only Once: Every glued contact removes two units of perimeter — one from the original shape and one from the new square.

Question 15

More questions in IT

Split and rejoin

Area Maze Puzzles