The perimeter of a semicircle is 6.682 kilometers. What is the semicircle's area?


To find the area of a semicircle when given its perimeter, we can follow these steps:

### Step 1: Express the perimeter of the semicircle.
The perimeter \( P \) of a semicircle consists of the straight edge (diameter) and the curved edge (half the circumference of a full circle):
\[
P = \pi r + 2r
\]
where:
- \( r \) is the radius of the semicircle,
- \( \pi r \) is the half-circumference,
- \( 2r \) is the diameter.

We are given that the perimeter \( P = 6.682 \) kilometers, so:
\[
\pi r + 2r = 6.682
\]

### Step 2: Solve for the radius \( r \).
We can factor out \( r \) from the equation:
\[
r(\pi + 2) = 6.682
\]

Now solve for \( r \):
\[
r = \frac{6.682}{\pi + 2}
\]
Substitute \( \pi \approx 3.1416 \):
\[
r = \frac{6.682}{3.1416 + 2} = \frac{6.682}{5.1416}
\]

Calculating \( r \):
\[
r \approx 1.299 \, \text{kilometers}
\]

### Step 3: Calculate the area of the semicircle.
The area \( A \) of a semicircle is given by:
\[
A = \frac{1}{2} \pi r^2
\]
Substitute the value of \( r \):
\[
A = \frac{1}{2} \times 3.1416 \times (1.299)^2
\]

Calculating \( A \):
\[
A \approx \frac{1}{2} \times 3.1416 \times 1.687
\]
\[
A \approx 2.649 \, \text{square kilometers}
\]

### Final Answer:
The area of the semicircle is approximately \( 2.649 \, \text{km}^2 \).

Would you like further details or have any questions?

Here are 5 related questions you might consider:
1. How do you calculate the area of a full circle given its radius?
2. How would the perimeter formula change for a quarter circle?
3. What is the relationship between the radius and the diameter of a circle?
4. Can the formula for the perimeter of a semicircle be applied to any sector of a circle?
5. How does the area of a semicircle compare to the area of a full circle?

**Tip:** Always double-check if the problem involves a portion of the circle (like a semicircle) to apply the correct formulas for area and perimeter!

The perimeter of a semicircle is 6.682 kilometers. What is the semicircle's area?

Learn how to calculate the area of a semicircle with a perimeter of 6.682 km. This problem covers essential geometry concepts such as the relationship between a circle's radius, perimeter, and area. The solution includes step-by-step calculations for the radius and area of the semicircle. Suitable for students in Grades 8-10.

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