Akshi says that the perimeter of this triangle shape is 9 units. Toshi says it can’t be 9 units and the perimeter will be more than 9 units. What do you think?

Question

Homi · Accepted Answer

Understand the Question

The triangle formed on the dot grid has two straight sides (one vertical, one horizontal) of length $3\text{ units}$ each.
The third side is diagonal. On a grid, a diagonal line connecting points is always strictly longer than the straight line covering the same number of grid units (longer than $3\text{ units}$ ).
Since the two straight sides already sum to $3 + 3 = 6\text{ units}$ , and the diagonal side is greater than $3\text{ units}$ , the total perimeter must be greater than $6 + 3 = 9\text{ units}$ .
Therefore, Toshi is correct and Akshi is incorrect.

Step 1 · Find Lengths of the Straight Sides

The distance between two consecutive horizontal or vertical dots is $1\text{ unit}$ . Diagram 1

\begin{aligned} \text{Length of vertical side} &= 3 \text{ units} \\ \text{Length of horizontal side} &= 3 \text{ units} \end{aligned}

Step 2 · Find Length of the Diagonal Side

The diagonal side connects the top-left and bottom-right points. Diagram 2

A diagonal distance across a grid is always longer than a straight distance of the same number of units ( $> 3\text{ units}$ ).

\begin{aligned} 3^2 + 3^2 &= 9 + 9 = 18 \\ \text{Length of diagonal side} &= \sqrt{18} \approx 4.24 \text{ units} \end{aligned}

Step 3 · Calculate the Total Perimeter

Perimeter is the sum of the lengths of all three sides: Diagram 3

\begin{aligned} \text{Perimeter} &= 3 \text{ units} + 3 \text{ units} + 4.24 \text{ units} \\ &= 6 \text{ units} + 4.24 \text{ units} \\ &= 10.24 \text{ units} \end{aligned}

$\text{Perimeter} \approx 10.24 \text{ units}$

Step 4 · Compare and Conclude

Akshi's claim: $\text{Perimeter} = 9 \text{ units}$ (Incorrect)
Toshi's claim: $\text{Perimeter} > 9 \text{ units}$ (Correct)

Since the diagonal side is longer than $3\text{ units}$ (approximately $4.24\text{ units}$ ), the total perimeter is approximately $10.24\text{ units}$ , which is greater than $9\text{ units}$ .

Answer

Toshi is correct. The perimeter is approximately $10.24\text{ units}$ , which is more than $9\text{ units}$ .

Common Mistakes

Treating Diagonal Units as Straight Units: Counting the diagonal side as $3\text{ units}$ by simply counting the dot intervals instead of recognizing that diagonal spacing on a square grid is $\sqrt{2} \approx 1.414$ times longer than straight spacing.
Confusing Perimeter with Area: Calculating the number of squares inside the shape rather than the boundary length around it.

Question 1

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