The ratio of the areas of the circumcircle and the incircle of a square is

Options

A. `2 : 1`
B. `1 : 2`
C. `a//2 : 1`
D. `1//a2`

Answer

To find the ratio of the areas of the circumcircle and the incircle of a square, we can follow these steps:

### Step 1: Understand the Shapes
A square has both a circumcircle (the circle that passes through all four vertices) and an incircle (the circle that is tangent to all four sides).

### Step 2: Define the Side Length
Let the side length of the square be denoted as \( s \).

### Step 3: Calculate the Area of the Incircle
The radius of the incircle is equal to half the side length of the square. Therefore, the radius \( r \) of the incircle is:
\[ r = \frac{s}{2} \]
The area \( A_{in} \) of the incircle is given by the formula:
\[ A_{in} = \pi r^2 = \pi \left(\frac{s}{2}\right)^2 = \pi \frac{s^2}{4} \]

### Step 4: Calculate the Area of the Circumcircle
The radius of the circumcircle is equal to half the diagonal of the square. The diagonal \( d \) of the square can be calculated using the Pythagorean theorem:
\[ d = s\sqrt{2} \]
Thus, the radius \( R \) of the circumcircle is:
\[ R = \frac{d}{2} = \frac{s\sqrt{2}}{2} \]
The area \( A_{circ} \) of the circumcircle is:
\[ A_{circ} = \pi R^2 = \pi \left(\frac{s\sqrt{2}}{2}\right)^2 = \pi \frac{2s^2}{4} = \pi \frac{s^2}{2} \]

### Step 5: Find the Ratio of the Areas
Now we can find the ratio of the area of the circumcircle to the area of the incircle:
\[ \text{Ratio} = \frac{A_{circ}}{A_{in}} = \frac{\pi \frac{s^2}{2}}{\pi \frac{s^2}{4}} = \frac{\frac{s^2}{2}}{\frac{s^2}{4}} \]
When we simplify this, the \( \pi \) and \( s^2 \) cancel out:
\[ \text{Ratio} = \frac{1/2}{1/4} = \frac{1}{2} \times \frac{4}{1} = 2 \]

### Step 6: Final Ratio
Thus, the ratio of the areas of the circumcircle to the incircle of a square is:
\[ \text{Ratio} = 2:1 \]

### Conclusion
The final answer is that the ratio of the areas of the circumcircle and the incircle of a square is \( 2:1 \).

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