To find the area of sector \( CDE \) in circle \( D \), we need to use the formula for the area of a sector:
\[ \text{Area of the sector} = \frac{\theta}{360} \times \pi r^2 \]
where \( \theta \) is the central angle in degrees and \( r \) is the radius of the circle.
-
Identify the values:
- Given \( m\angle CDE = 144^\circ \)
- The distance \( CD = 4 \). This is the radius \( r \) of the circle, so \( r = 4 \).
-
Plug the values into the formula: \[ \text{Area of the sector} = \frac{144}{360} \times \pi (4^2) \]
-
Calculate \( r^2 \): \[ 4^2 = 16 \]
-
Calculate the fraction: \[ \frac{144}{360} = \frac{2}{5} \text{ (simplifying 144 and 360)} \]
-
Substituting the values: \[ \text{Area of the sector} = \frac{2}{5} \times \pi \times 16 \] \[ = \frac{32\pi}{5} \]
-
Calculate this value: \[ \text{Area of the sector} \approx \frac{32 \times 3.14159}{5} \approx \frac{100.53088}{5} \approx 20.106176 \]
-
Round to the nearest hundredth: \[ \text{Area of the sector} \approx 20.11 \]
Thus, the area of sector \( CDE \) is approximately 20.11 square units.