ABCD is a square inscribed in a circle of radius r. Then the total area (in square units) of the portions of the circle lying outside the square is

Options

A. `pi(r^(2) - 4)`
B. `2pi(r^(2) -1)`
C. `pi^(2) r(r-7)`
D. `r^(2)(pi - 2)`

Answer

To solve the problem, we need to find the area of the portions of the circle that lie outside the inscribed square ABCD. Here are the steps to arrive at the solution:

### Step 1: Calculate the area of the circle.
The area \( A_{circle} \) of a circle is given by the formula:
\[ A_{circle} = \pi r^2 \]
where \( r \) is the radius of the circle.

### Step 2: Determine the side length of the inscribed square.
For a square inscribed in a circle, the diagonal of the square is equal to the diameter of the circle. The diameter \( D \) of the circle is:
\[ D = 2r \]
If \( s \) is the side length of the square, then the diagonal \( d \) of the square can be expressed using the Pythagorean theorem:
\[ d = s\sqrt{2} \]
Setting the diagonal equal to the diameter, we have:
\[ s\sqrt{2} = 2r \]
From this, we can solve for \( s \):
\[ s = \frac{2r}{\sqrt{2}} = r\sqrt{2} \]

### Step 3: Calculate the area of the square.
The area \( A_{square} \) of the square is given by:
\[ A_{square} = s^2 = (r\sqrt{2})^2 = 2r^2 \]

### Step 4: Calculate the area outside the square.
The area outside the square is the area of the circle minus the area of the square:
\[ A_{outside} = A_{circle} - A_{square} \]
Substituting the areas we calculated:
\[ A_{outside} = \pi r^2 - 2r^2 \]
This can be simplified to:
\[ A_{outside} = (\pi - 2)r^2 \]

### Final Answer:
Thus, the total area of the portions of the circle lying outside the square is:
\[ \boxed{(\pi - 2)r^2} \]

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