Halving a Square Using Paper

Cut out a square from a piece of paper. Now make a square whose area is half the area of the first square.

Question

Halving a Square Using Paper Cut out a square from a piece of paper. Now make a square whose area is half the area of the first square.

Homi · Accepted Answer

Understand the Question

The area of a square of side length $s$ is $A_1 = s^2$ .
By folding each of the four corners of the paper square inwards so that their vertices meet at the center, we form a new inner square connecting the midpoints of the original sides.
The four folded right-angled triangles account for half of the original square's total area, leaving the area of the newly formed square as $\dfrac{s^2}{2}$ , which is exactly half of the initial area.

Step 1 · Prepare the Original Square and Find Midpoints

Let the side length of the original square be $s$ . $A_1 = s \times s = s^2$

Fold the square in half horizontally and vertically, then unfold to locate the center and the midpoints of all four sides. Diagram 1

Step 2 · Fold the Corners Inwards

Fold each of the four corners inwards so that each corner vertex meets exactly at the center of the square. Diagram 2

The fold lines join the adjacent midpoints to form a new inner square with area $A_2$ .

Step 3 · Calculate the Area of the New Square

Folding the four corners forms four identical right-angled triangles, each having legs of length $\dfrac{s}{2}$ .

\begin{aligned} \text{Area of one triangle} &= \dfrac{1}{2} \times \text{base} \times \text{height} \\[0.6em] &= \dfrac{1}{2} \times \dfrac{s}{2} \times \dfrac{s}{2} \\[0.6em] &= \dfrac{s^2}{8} \end{aligned}

\begin{aligned} \text{Total area removed} &= 4 \times \dfrac{s^2}{8} \\[0.6em] &= \dfrac{4s^2}{8} \\[0.6em] &= \dfrac{s^2}{2} \end{aligned}

\begin{aligned} A_2 &= A_1 - \text{Total area removed} \\[0.6em] &= s^2 - \dfrac{s^2}{2} \\[0.6em] &= \dfrac{s^2}{2} \end{aligned}

Thus, the area of the new square is half the area of the first square.

Answer

Fold each of the four corners of the square inwards to meet at its center. The resulting square has area $\dfrac{s^2}{2}$ , which is half the area of the original square.

Common Mistakes

Halving Side Length Instead of Area: Halving the side length from $s$ to $\dfrac{s}{2}$ reduces the area to $\left(\dfrac{s}{2}\right)^2 = \dfrac{s^2}{4}$ (one-fourth), not $\dfrac{1}{2}$ .
Folding Edge-to-Edge: Folding a square along a centerline gives a rectangle of half the area, not a square. Folding all four corners inwards to the center is required to form a new square.

Question 2

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