Question

A square paper of side 10 cm is folded sequentially and a circular cut of radius 1 cm is made at the corners as shown below. After unfolding the sheet completely, what will be the total area of all the pieces which have been cut-out from the original square sheet? Assume the value of $\pi$ = 3.14.

Hint:

Instead of counting fabric layers, think of each fold as a mirror line. Unfolding the cut corner means reflecting that same quarter-circle piece across each fold line in turn, and every 4 quarter-circles you collect this way make up one full circle.

Answer 1

Correct Answer: 2

Step 1: Understanding the Question:
A square piece of paper is folded multiple times.
At the corners of the final folded paper, circular cuts are made with a radius of $1\text{ cm}$.
We need to calculate the total area of all the small pieces that are completely cut out and removed from the original sheet after unfolding.

Step 2: Key Formula or Approach:
We analyze the folding sequence to find how many layers of paper are cut:
1. Folding in half vertically once divides the sheet into 2 layers.
2. Folding in half horizontally once more divides the sheet into 4 layers.
3. Folding horizontally one last time creates a total of 8 layers.
The area of a sector cut out at any corner is a quarter of a circle because all corner angles of a rectangle are $90^{\circ}$.
Total Cut Area = (Number of Layers) $\times$ (Total Cut Area on one layer).

Step 3: Detailed Explanation:
1. Analyze the Folds:
- Start with a $10\text{ cm} \times 10\text{ cm}$ square.
- First Fold: Crease is along the vertical centerline. Dimensions become $5\text{ cm} \times 10\text{ cm}$ (2 layers).
- Second Fold: Crease is along the horizontal centerline. Dimensions become $5\text{ cm} \times 5\text{ cm}$ (4 layers).
- Third Fold: Crease is along a horizontal line again. Dimensions become $5\text{ cm} \times 2.5\text{ cm}$ (8 layers).
2. Analyze the Cuts:
- The final folded sheet is a rectangle of dimensions $5\text{ cm} \times 2.5\text{ cm}$.
- At each of the 4 corners of this folded rectangle, a circular cut of radius $R = 1\text{ cm}$ is made.
- Because each corner of a rectangle contains a right angle ($90^{\circ}$), each cut removes exactly a quarter-circle from each layer at that corner.
- Since there are 4 corners, the total area removed from one layer of the folded sheet is:
\[ A_{\text{layer}} = 4 \times \left( \frac{1}{4} \pi R^2 \right) = \pi R^2 \]
3. Calculating Total Removed Area:
- Since there are 8 layers of paper stacked together, cutting through the 4 corners removes this area from all 8 layers:
\[ A_{\text{total}} = 8 \times A_{\text{layer}} = 8 \times \pi R^2 \]
- Substituting $R = 1\text{ cm}$ and $\pi = 3.14$:
\[ A_{\text{total}} = 8 \times 3.14 \times (1)^2 = 25.12\text{ cm}^2 \]

Step 4: Final Answer:
The total area of all the cut-out pieces is 25.12 $\text{cm}^2$.

Answer 2

Concept:

• Unfolding a sheet of paper is the same as reflecting the cut shape across each fold line in turn. Every reflection doubles the number of cut-out pieces without changing the shape or area of any single piece.
• A quarter-circle cut at a corner of the folded sheet turns into more quarter-circles every time it is reflected across another fold line, and 4 quarter-circles always make up exactly one full circle.

Step 1: Track the folds as mirror lines.
The sheet is folded 3 times in total: once down the middle vertically, taking 10 cm to 5 cm, then twice more in half, taking 5 cm to 2.5 cm, giving a final folded rectangle of 5 cm by 2.5 cm made of 8 stacked layers. Each of these 3 folds becomes a straight mirror line once the sheet is opened back up.

Step 2: Follow one corner cut through all 3 reflections.
A quarter-circle of radius 1 cm is cut at a corner of the small folded rectangle. Reflecting it across the first fold line turns it into a half circle, made of 2 quarter-circles. Reflecting that across the second fold line turns it into a full circle, made of 4 quarter-circles. Reflecting that full circle across the third fold line gives 2 full circles side by side, made of 8 quarter-circles in total.

Step 3: Repeat for all 4 corners of the folded rectangle.
The folded rectangle has 4 corners, and Step 2 shows each one unfolds into 8 quarter-circles of radius 1 cm.
Total quarter-circles removed $=4\times8=32$.

Step 4: Convert the quarter-circles into a total area.
Every 4 quarter-circles make up one full circle, so 32 quarter-circles make up $32\div4=8$ full circles.
Total area removed $=8\times\pi(1)^2=8\times3.14=25.12$ cm$^2$.

Final Answer: 25.12 cm$^2$