Mark the sides of a square into thirds and cut off each of its corners as far as the marks. What shape is left?

Question

Homi · Accepted Answer

Understand the Question

Dividing each of the $4$ sides of a square into thirds marks $2$ points on each side.
Cutting off each corner "as far as the marks" removes a small right-angled triangle from each of the $4$ corners.
The remaining shape retains the middle third of each original side ( $4$ edges) and gains a new straight edge at each cut corner ( $4$ edges), resulting in an $8$ -sided polygon.

Step 1 · Divide Sides and Cut Corners

Let the square have a side length of $3\text{ units}$ . Dividing each side into three equal parts gives segments of length: $\dfrac{3\text{ units}}{3} = 1\text{ unit}$ Diagram 1

Cutting off each corner as far as the marks connects the first mark on one side to the adjacent mark on the perpendicular side, removing $4$ corner triangles. Diagram 2

Step 2 · Count Sides and Identify the Shape

Each corner cut introduces $1$ new straight edge, and the middle segment of each original side remains:

\begin{aligned} \text{Number of original sides (middle segments)} &= 4 \\ \text{Number of new sides created by cuts} &= 4 \\ \text{Total number of sides in the new shape} &= 4 + 4 = 8 \end{aligned}

A closed polygon with $8$ straight sides is called an octagon.

Answer

Octagon

Common Mistakes

Assuming it is a Regular Octagon: While the shape has $8$ sides (an octagon), the straight edges have length $1$ and the cut diagonal edges have length $\sqrt{1^2 + 1^2} = \sqrt{2}$ , so the sides are not all equal.
Under-counting Sides: Forgetting that cutting a corner replaces $1$ vertex with $1$ new flat side instead of reducing the total number of sides.

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