A regular octagon is formed by cutting an isosceles right triangle from each of the corners of a square with side 15 cm. The area (in `cm^2` ) of the octagon is

Options

A. `(30)/( sqrt(2) +1)`
B. `(450)/( sqrt(2) -1)`
C. `(30)/( 1- sqrt(2) )`
D. `(450)/( sqrt(2) +1)`

Answer

To find the area of the regular octagon formed by cutting isosceles right triangles from each corner of a square with a side length of 15 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Calculate the area of the square:**
The area of a square is given by the formula:
\[ \text{Area of square} = \text{side}^2 \]
Here, the side of the square is 15 cm. Therefore,
\[ \text{Area of square} = 15 \times 15 = 225 \, \text{cm}^2 \]

**Hint:** Remember that the area of a square is calculated by squaring the length of one of its sides.

2. **Determine the area of one isosceles right triangle:**
Let the legs of each isosceles right triangle be of length \( x \). The area of one triangle can be calculated using the formula:
\[ \text{Area of triangle} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times x \times x = \frac{x^2}{2} \]

**Hint:** An isosceles right triangle has two equal sides, which serve as the base and height.

3. **Calculate the total area of the four triangles:**
Since there are four corners, we multiply the area of one triangle by 4:
\[ \text{Total area of triangles} = 4 \times \frac{x^2}{2} = 2x^2 \]

**Hint:** When calculating the total area of multiple identical shapes, multiply the area of one shape by the number of shapes.

4. **Express the area of the octagon:**
The area of the octagon can be found by subtracting the total area of the triangles from the area of the square:
\[ \text{Area of octagon} = \text{Area of square} - \text{Total area of triangles} \]
Substituting the values we have:
\[ \text{Area of octagon} = 225 - 2x^2 \]

**Hint:** The area of the remaining shape (octagon) is found by removing the area of the cut-out triangles from the area of the original square.

5. **Relate \( x \) to the side of the octagon:**
The side of the octagon can be expressed in terms of \( x \) using the Pythagorean theorem. The length of the side of the octagon is \( 15 - 2x \) (since \( x \) is cut from both ends). The hypotenuse of the triangle (which is also the side of the octagon) is:
\[ \text{Hypotenuse} = x\sqrt{2} \]
Setting this equal to the side of the octagon:
\[ 15 - 2x = x\sqrt{2} \]

**Hint:** The relationship between the sides of the triangles and the octagon can be established using the Pythagorean theorem.

6. **Solve for \( x \):**
Rearranging the equation gives:
\[ 15 = x\sqrt{2} + 2x \]
Factoring out \( x \):
\[ 15 = x(\sqrt{2} + 2) \]
Therefore,
\[ x = \frac{15}{\sqrt{2} + 2} \]

**Hint:** Isolate \( x \) to find its value.

7. **Substitute \( x \) back to find the area of the octagon:**
Substitute \( x \) back into the area formula:
\[ \text{Area of octagon} = 225 - 2\left(\frac{15}{\sqrt{2} + 2}\right)^2 \]

**Hint:** Substitute carefully and simplify to find the final area.

8. **Calculate the final area:**
After substituting and simplifying, we find:
\[ \text{Area of octagon} = 225 - 2\left(\frac{225}{(\sqrt{2} + 2)^2}\right) \]
Simplifying further will yield the area of the octagon.

**Hint:** Use algebraic identities to simplify expressions when substituting values.

### Final Result:
The area of the octagon is approximately \( 225 - 2 \times \frac{225}{6} = 225 - 75 = 150 \, \text{cm}^2 \).