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We are given a circle with a radius of 14 cm and a sector with an arc length of 20 cm. The task is to find the area of the shaded region.

### Steps to solve the problem:

1. **Find the circumference of the entire circle**:
   The circumference \(C\) of a circle is given by:
   \[
   C = 2 \pi r
   \]
   where \(r = 14 \, \text{cm}\). Substituting this:
   \[
   C = 2 \pi \times 14 = 28 \pi \, \text{cm}
   \]

2. **Calculate the fraction of the circle corresponding to the sector**:
   The length of the arc of the sector is given as 20 cm. The fraction of the circle that this sector represents is:
   \[
   \text{Fraction of circle} = \frac{\text{Arc Length}}{\text{Circumference}} = \frac{20}{28 \pi}
   \]
   Simplifying:
   \[
   \text{Fraction of circle} = \frac{20}{28 \pi} = \frac{5}{7 \pi}
   \]

3. **Find the area of the entire circle**:
   The area \(A\) of a circle is given by:
   \[
   A = \pi r^2
   \]
   Substituting \(r = 14\) cm:
   \[
   A = \pi \times 14^2 = \pi \times 196 = 196 \pi \, \text{cm}^2
   \]

4. **Find the area of the sector**:
   The area of the sector is the fraction of the circle's area, which is:
   \[
   \text{Area of sector} = \left( \frac{5}{7 \pi} \right) \times 196 \pi = \frac{5 \times 196}{7} = 140 \, \text{cm}^2
   \]

Thus, the area of the shaded region is **140 cm²**.

Would you like more details or have any questions? Here are some related questions:

1. How do you calculate the area of a sector in radians?
2. What other formulas can be used to find the arc length of a sector?
3. How would the result change if the radius of the circle were different?
4. What is the relationship between the angle of the sector and its arc length?
5. How can the area of the sector be calculated using degrees instead of arc length?

**Tip:** Always remember that the arc length is proportional to the fraction of the circle that the sector represents!

The radius of the circle is 14 cm. The arc length of the sector is 20 cm. Find the area of the shaded region.

Learn how to find the area of a shaded sector in a circle with a radius of 14 cm and an arc length of 20 cm. This problem covers key geometry concepts such as the circumference of a circle, sector areas, and arc length proportionality. Suitable for students in Grades 9-10.

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