The ratio of the area of the incircle and the circum-circle of a square is

Options

A. `1:2`
B. `sqrt(2):1`
C. `1:sqrt(2)`
D. `2:1`

Answer

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

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

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

### Step 3: Calculate the area of the incircle
The area of a circle is given by the formula \( A = \pi r^2 \). Substituting the radius of the incircle:
\[ \text{Area of incircle} = \pi \left(\frac{s}{2}\right)^2 = \pi \frac{s^2}{4} \]

### Step 4: Calculate the radius of the circumcircle
The radius of the circumcircle (R) of a square is equal to half 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:
\[ \text{Area of circumcircle} = \pi R^2 = \pi \left(\frac{s\sqrt{2}}{2}\right)^2 = \pi \frac{2s^2}{4} = \frac{\pi s^2}{2} \]

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

### Final Answer
The ratio of the area of the incircle to the area of the circumcircle of a square is:
\[ \frac{1}{2} \]

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