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find the area of the shaded region (four decimal places)

To find the area of the shaded region, we'll start by calculating the area of the entire sector and then focus on the shaded part.

### Step 1: Calculate the Area of the Whole Sector
The formula for the area of a sector is:

\[
\text{Sector Area} = \left(\frac{\text{angle of sector}}{360}\right) \times \pi r^2
\]

Given:
- Angle of the sector = \(48^\circ\)
- Radius \(r = 14 \text{ cm}\)

Substitute the values into the formula:

\[
\text{Sector Area} = \left(\frac{48}{360}\right) \times \pi \times (14)^2
\]

\[
\text{Sector Area} = \left(\frac{48}{360}\right) \times \pi \times 196
\]

\[
\text{Sector Area} = \left(\frac{48}{360}\right) \times 615.7522 \text{ cm}^2 \quad (\text{using } \pi \approx 3.1416)
\]

\[
\text{Sector Area} = \left(\frac{48}{360}\right) \times 615.7522
\]

\[
\text{Sector Area} = 0.1333 \times 615.7522 = 82.1003 \text{ cm}^2
\]

So, the area of the whole sector is **82.1003 cm²** (rounded to four decimal places).

### Step 2: Determine the Area of the Shaded Region
Since the problem specifies finding the area of the shaded region, which appears to be part of the sector, the area we've just calculated is already for the specific sector. If the shaded part refers only to this sector, then the shaded region area is 82.1003 cm².

Would you like more details or have any questions about this? Here are some related questions that could help deepen your understanding:

1. How do we calculate the area if the shaded region is a segment and not the entire sector?
2. What changes if the angle of the sector is different?
3. How do you find the perimeter of the sector?
4. What happens to the sector area if the radius is doubled?
5. How would you calculate the area of a sector if only the arc length is known?

**Tip:** Always double-check if the shaded region refers to a sector, a segment, or another specific part of a circle before finalizing your answer.

Learn how to calculate the area of the shaded region in a circle sector using the sector area formula. This problem involves finding the area of a sector with a given angle and radius, suitable for advanced high school students.

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