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

The perimeter of a shape is the total length of its outer boundary.

Step 1 — Find the original perimeter

Let us call each side of a small square 1 unit. The original shape is a plus sign. It is made of 5 squares. Let us count the exposed sides of the squares. The central square has no exposed sides. Each of the 4 outer squares has 3 exposed sides. So, the total perimeter is the sum of these sides.

$3 \text{ units} + 3 \text{ units} + 3 \text{ units} + 3 \text{ units}$

$= 4 \times 3 \text{ units}$

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

Diagram 1

Step 2 — How perimeter changes

When we add a new square, it has 4 sides. If the new square touches the shape, some sides hide. Each side that touches removes 2 units. One unit is from the new square's side. One unit is from the existing shape's side. The new square adds its un-touched sides.

Step 3 — (a) Increase the perimeter

To increase perimeter, the new square touches 1 side. This means 3 sides of the new square are added. And 1 side of the old shape is removed. The perimeter increases by 2 units. Let us place the new square above the top square. This new square touches only 1 side. The new perimeter is the original plus 2 units.

$12 \text{ units} + 2 \text{ units}$

$\boxed{14 \text{ units}}$

Diagram 2

Step 4 — (b) Decrease the perimeter

To decrease perimeter, the new square touches 3 sides. If it touches 3 sides, 1 new side is added. And 3 sides of the old shape are removed. Change = $1 - 3 = \textbf{-2 units}$ . The plus sign has no "inner corners" or "holes". So, a new square cannot touch 3 sides. Therefore, it is not possible to decrease the perimeter.

Diagram 3

Step 5 — (c) Keep the perimeter the same

To keep perimeter same, the new square touches 2 sides. This means 2 sides of the new square are added. And 2 sides of the old shape are removed. The perimeter changes by 0 units. Let us place the new square in the bottom-right. Look at the example in the original image. The arrow shows a square placed here. This new square touches 2 sides. The new perimeter is the original plus 0 units.

$12 \text{ units} + 0 \text{ units}$

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

Diagram 4

Answer

(a) The perimeter increases to 14 units. (b) It is not possible to decrease the perimeter for this shape. (c) The perimeter stays the same at 12 units.

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