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

When we cut off the corners of a square, we create a new shape with more sides.

Step 1 — Setting up the square

Let us imagine a square. Let its side length be 3 units. We divide each side into three equal parts. So, each small part will have a length of $3 \text{ units} \div 3 = \mathbf{1 \text{ unit}}$ .

Diagram 1

Step 2 — Making the cuts

We cut off each corner "as far as the marks". This means we connect the first mark on one side to the first mark on the adjacent side. For example, at corner A, we connect the mark on side AB (let's call it P1) to the mark on side AD (let's call it S2). This cut removes a small triangle from each corner. Each cut forms a new side for the remaining shape.

Diagram 2

Step 3 — Counting the sides

The original square has 4 sides. When we cut off a corner, we remove a small triangle. This cut creates one new side for the shape that is left. Since there are 4 corners, we make 4 such cuts. Each cut adds one side to the total count. The middle part of each original side also remains. So, we have 4 segments from the original sides and 4 new segments from the cuts.

$\text{Number of original sides} = 4$

$\text{Number of new sides created by cuts} = 4$

$\text{Total number of sides in the new shape} = 4 + 4$

$\boxed{\text{Total sides} = 8}$

Step 4 — Identifying the shape

A polygon is a closed shape with straight sides. The number of sides tells us the name of the polygon. A polygon with 8 sides is called an octagon. The shape left after cutting the corners is an octagon.

Answer

The shape left is an octagon.

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