To find the area of a sector bounded by a 90-degree arc within a circle with a diameter of 4 centimeters, follow these steps:
1. Calculate the radius of the circle:
The diameter of the circle is 4 centimeters. Since the radius is half of the diameter,
[tex]\[
\text{radius} = \frac{\text{diameter}}{2} = \frac{4 \text{ cm}}{2} = 2 \text{ cm}
\][/tex]
2. Convert the angle from degrees to radians:
Angles in degrees can be converted to radians using the formula:
[tex]\[
\text{angle in radians} = \text{angle in degrees} \times \left(\frac{\pi}{180}\right)
\][/tex]
For a 90-degree angle,
[tex]\[
\text{angle in radians} = 90 \times \left(\frac{\pi}{180}\right) = \frac{\pi}{2}
\][/tex]
3. Use the formula for the area of a sector:
The formula to calculate the area of a sector is:
[tex]\[
\text{area of sector} = \frac{1}{2} \times \text{radius}^2 \times \text{angle in radians}
\][/tex]
Plugging in the values we have:
[tex]\[
\text{area of sector} = \frac{1}{2} \times (2 \text{ cm})^2 \times \frac{\pi}{2}
\][/tex]
4. Simplify the expression:
[tex]\[
\text{area of sector} = \frac{1}{2} \times 4 \text{ cm}^2 \times \frac{\pi}{2} = 2 \text{ cm}^2 \times \frac{\pi}{2}
\][/tex]
[tex]\[
\text{area of sector} = \pi \text{ cm}^2
\][/tex]
Thus, the exact area of the sector bounded by a 90-degree arc in a circle with a diameter of 4 centimeters is:
[tex]\[
\pi \text{ square centimeters}
\][/tex]
