The angle of a sector of a circle is `90^(@)`. The ratio of the area of the sector and that of the circle is ______.

Options

A. `4:1`
B. `3:1`
C. `1:4`
D. `3:1`

Explanation

To find the ratio of the area of a sector of a circle to the area of the entire circle when the angle of the sector is \(90^\circ\), we can follow these steps:

### Step 1: Understand the formulas for area
The area of a circle is given by the formula:
\[ \text{Area of Circle} = \pi r^2 \]
where \(r\) is the radius of the circle.

### Step 2: Calculate the area of the sector
The area of a sector of a circle can be calculated using the formula:
\[ \text{Area of Sector} = \frac{\theta}{360} \times \pi r^2 \]
where \(\theta\) is the angle of the sector in degrees.

### Step 3: Substitute the values
In this case, the angle \(\theta\) is \(90^\circ\). So, we substitute \(\theta\) into the formula for the area of the sector:
\[ \text{Area of Sector} = \frac{90}{360} \times \pi r^2 \]

### Step 4: Simplify the expression
Now, simplify the fraction:
\[ \frac{90}{360} = \frac{1}{4} \]
Thus, the area of the sector becomes:
\[ \text{Area of Sector} = \frac{1}{4} \times \pi r^2 \]

### Step 5: Find the ratio of the areas
Now, we need to find the ratio of the area of the sector to the area of the circle:
\[ \text{Ratio} = \frac{\text{Area of Sector}}{\text{Area of Circle}} = \frac{\frac{1}{4} \pi r^2}{\pi r^2} \]

### Step 6: Simplify the ratio
When we simplify this ratio, we notice that \(\pi r^2\) cancels out:
\[ \text{Ratio} = \frac{1}{4} \]

### Final Answer
Therefore, the ratio of the area of the sector to the area of the circle is:
\[ \text{Ratio} = \frac{1}{4} \]

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