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:

Unfolding can be modeled as geometric reflections. Reflections conserve area, so the total area of the cut-out shapes is simply the area cut out from a single layer multiplied by the total number of layers.

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 $\text{cm}^2$.