To determine the area of the semicircle with a given perimeter of 5.14 millimeters, we can follow these steps:
1. Understanding the Perimeter of a Semicircle:
The perimeter (or circumference) of a semicircle formula includes the straight-line diameter and the curved edge of the semicircle. It is given by:
[tex]\[
\text{Perimeter} = \pi r + 2r
\][/tex]
where [tex]\( r \)[/tex] is the radius of the semicircle.
2. Solving for the Radius:
Given the perimeter of 5.14 millimeters:
[tex]\[
5.14 = \pi r + 2r
\][/tex]
Factor [tex]\( r \)[/tex] out of the equation:
[tex]\[
5.14 = r(\pi + 2)
\][/tex]
Solve for [tex]\( r \)[/tex]:
[tex]\[
r = \frac{5.14}{\pi + 2}
\][/tex]
3. Using the Given Value of [tex]\(\pi\)[/tex]:
Given [tex]\(\pi \approx 3.14\)[/tex], substitute this into the equation:
[tex]\[
r = \frac{5.14}{3.14 + 2} = \frac{5.14}{5.14} \approx 1.00 \, \text{millimeters}
\][/tex]
4. Calculating the Area of the Semicircle:
The area of a semicircle can be calculated using the formula:
[tex]\[
\text{Area} = \frac{1}{2} \pi r^2
\][/tex]
Substitute [tex]\( r = 1.00 \, \text{millimeters} \)[/tex] and [tex]\(\pi \approx 3.14\)[/tex] into the formula:
[tex]\[
\text{Area} = \frac{1}{2} \times 3.14 \times (1.00)^2
= \frac{1}{2} \times 3.14 \times 1.00
= 1.57 \, \text{square millimeters}
\][/tex]
5. Rounding:
The area is already rounded to the nearest hundredth place.
Therefore, the area of the semicircle is:
[tex]\[
\boxed{1.57} \, \text{square millimeters}
\][/tex]
