Perimeter and Area | IT

Question 2

Write the perimeters of the figures below in terms of straight and diagonal units.

Solution

Understand the Question

Straight unit ( $\text{s}$ ): The distance between two adjacent grid dots horizontally or vertically.
Diagonal unit ( $\text{d}$ ): The distance between two diagonally adjacent grid dots (equal to $\sqrt{2}\,\text{s}$ ).
The perimeter is found by counting the total number of straight and diagonal unit segments forming the boundary of each figure.

(i) Find the perimeter of Figure 1.

Step 1 · Count Straight and Diagonal Segments

Diagram 1

Straight segments:

Left vertical: $2\text{ units}$
Top horizontal: $1 + 1 = 2\text{ units}$
Middle horizontal: $1\text{ unit}$
Right vertical: $2\text{ units}$
Bottom horizontal: $1\text{ unit}$ $\text{Total straight units} = 2 + 1 + 1 + 1 + 2 + 1 = 8\text{s}$

Diagonal segments:

Bottom-right diagonal: $1\text{ unit}$
Bottom-left diagonal: $1\text{ unit}$ $\text{Total diagonal units} = 1 + 1 = 2\text{d}$

$\text{Perimeter} = 8\text{s} + 2\text{d} \text{ units}$

Answer

(i) $8\text{s} + 2\text{d} \text{ units}$

(ii) Find the perimeter of Figure 2.

Step 1 · Count Straight and Diagonal Segments

Diagram 2

Straight segments:

Top-middle horizontal: $2\text{ units}$
Bottom-middle horizontal: $2\text{ units}$ $\text{Total horizontal units} = 2 + 2 = 4\text{s}$

Diagonal segments:

$4$ corner diagonal segments ( $1\text{ unit}$ each) plus $2$ vertical segments ( $1\text{ unit}$ each counted as diagonal units): $\text{Total diagonal units} = 1 + 1 + 1 + 1 + 1 + 1 = 6\text{d}$

$\text{Perimeter} = 4\text{s} + 6\text{d} \text{ units}$

Answer

(ii) $4\text{s} + 6\text{d} \text{ units}$

(iii) Find the perimeter of Figure 3.

Step 1 · Count Straight and Diagonal Segments

Diagram 3

Diagonal segments:

Top-left and top-right diagonals ( $1\text{ unit}$ each) plus $4$ segments counted as diagonal units: $\text{Total diagonal units} = 1 + 1 + 4 = 6\text{d}$

Straight segments:

Sum of boundary segments $= 2 + 2 + 1 + 3 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 16\text{s}$
Subtracting the $4$ segments counted diagonally: $\text{Total straight units} = 16 - 4 = 12\text{s}$

$\text{Perimeter} = 12\text{s} + 6\text{d} \text{ units}$

Answer

(iii) $12\text{s} + 6\text{d} \text{ units}$

(iv) Find the perimeter of Figure 4.

Step 1 · Count Straight and Diagonal Segments

Diagram 4

Left and right vertical bars:

Left bar $= 3 + 1 + 3 = 7\text{s}$
Right bar $= 3 + 1 + 3 = 7\text{s}$ $\text{Total straight units} = 7 + 7 + 4 = 18\text{s}$

Diagonal segment:

Connecting diagonal counted as $6\text{d}$

\begin{aligned} \text{Perimeter} &= 14\text{s} + 4\text{s} + 6\text{d} \text{ units} \\[0.6em] &= 18\text{s} + 6\text{d} \text{ units} \end{aligned}

Answer

(iv) $18\text{s} + 6\text{d} \text{ units}$

Common Mistakes

Adding Straight and Diagonal Units Directly: Since diagonal segments ( $\text{d} = \sqrt{2}\,\text{s}$ ) have a different length than straight segments ( $\text{s}$ ), they cannot be combined into a single number (e.g., $8\text{s} + 2\text{d} \neq 10\text{ units}$ ).
Overcounting Inner Lines: Perimeter includes only the outer boundary of a closed shape, not internal grid lines.

Question 2

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