What is the side of the largest possible regular octagon that can be cut out of a square of side 1 cm?

Options

A. `sqrt2-1`cm
B. `(1)/(sqrt(2)+2)cm`
C. `(1)/(sqrt(2)-1)cm`
D. `(sqrt(2))/(sqrt(2)+1)cm`

Answer

To find the side of the largest possible regular octagon that can be cut out of a square with a side length of 1 cm, we can follow these steps:

### Step 1: Understand the Geometry
We start with a square of side length \( a = 1 \) cm. We need to inscribe a regular octagon within this square.

### Step 2: Define the Side Length of the Octagon
Let the side length of the octagon be \( x \). The octagon can be visualized as being formed by cutting off the corners of the square.

### Step 3: Analyze the Right Triangle Formed
When we cut off the corners of the square, we form right triangles at each corner. Each triangle has angles of 45 degrees (since the octagon is regular), and the legs of the triangle will be equal in length.

Let \( PQ \) and \( QR \) be the legs of one of these right triangles. Since the angles are 45 degrees, we have:
\[ PQ = QR \]

### Step 4: Apply the Pythagorean Theorem
According to the Pythagorean theorem:
\[ PQ^2 + QR^2 = PR^2 \]
Since \( PQ = QR \), we can denote \( PQ = QR = y \). Thus, we can write:
\[ 2y^2 = x^2 \]
This implies:
\[ y^2 = \frac{x^2}{2} \]
So:
\[ y = \frac{x}{\sqrt{2}} \]

### Step 5: Relate the Side Lengths
The total length of the square's side can be expressed as:
\[ a = x + y + y \]
Substituting \( y \):
\[ a = x + \frac{x}{\sqrt{2}} + \frac{x}{\sqrt{2}} \]
This simplifies to:
\[ a = x + \frac{2x}{\sqrt{2}} \]
\[ a = x \left( 1 + \sqrt{2} \right) \]

### Step 6: Solve for \( x \)
Now, since \( a = 1 \) cm, we can set up the equation:
\[ 1 = x(1 + \sqrt{2}) \]
Solving for \( x \):
\[ x = \frac{1}{1 + \sqrt{2}} \]

### Step 7: Rationalize the Denominator
To simplify \( x \), we rationalize the denominator:
\[ x = \frac{1(1 - \sqrt{2})}{(1 + \sqrt{2})(1 - \sqrt{2})} \]
This gives:
\[ x = \frac{1 - \sqrt{2}}{1 - 2} \]
\[ x = 1 - \sqrt{2} \]

### Step 8: Final Calculation
Thus, we have:
\[ x = \sqrt{2} - 1 \text{ cm} \]

### Conclusion
The side length of the largest possible regular octagon that can be cut out of a square of side 1 cm is:
\[ x = \sqrt{2} - 1 \text{ cm} \]

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