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

Options

A. ` 1 : 2`
B. ` 1 : 3`
C. ` 2 : :1`
D. None of these

Answer

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

### Step 1: Define the Square
Let the side length of the square be denoted as \( s \).

### Step 2: Determine the Radius of the Incircle
The radius of the incircle (\( r \)) of a square is equal to half the length of the side of the square. Therefore, we have:
\[ r = \frac{s}{2} \]

### Step 3: Calculate the Area of the Incircle
The area of the incircle can be calculated using the formula for the area of a circle, which is \( \pi r^2 \):
\[ \text{Area of incircle} = \pi r^2 = \pi \left(\frac{s}{2}\right)^2 = \pi \frac{s^2}{4} \]

### Step 4: Determine the Radius of the Circumcircle
The radius of the circumcircle (\( R \)) of a square is equal to half the length of the diagonal of the square. The diagonal \( d \) of a square can be calculated using the Pythagorean theorem:
\[ d = s\sqrt{2} \]
Thus, the radius of the circumcircle is:
\[ R = \frac{d}{2} = \frac{s\sqrt{2}}{2} \]

### Step 5: Calculate the Area of the Circumcircle
Using the formula for the area of a circle, the area of the circumcircle is:
\[ \text{Area of circumcircle} = \pi R^2 = \pi \left(\frac{s\sqrt{2}}{2}\right)^2 = \pi \frac{2s^2}{4} = \frac{\pi s^2}{2} \]

### Step 6: Find the Ratio of the Areas
Now, we can find the ratio of the area of the incircle to the area of the circumcircle:
\[ \text{Ratio} = \frac{\text{Area of incircle}}{\text{Area of circumcircle}} = \frac{\pi \frac{s^2}{4}}{\frac{\pi s^2}{2}} = \frac{\frac{s^2}{4}}{\frac{s^2}{2}} = \frac{1/4}{1/2} = \frac{1}{2} \]

### Final Answer
Thus, the ratio of the areas of the incircle and circumcircle of a square is:
\[ \boxed{1:2} \]