To find the area of a circle given its circumference, we need to follow these steps:
1. Understand the relationship between circumference and radius:
The circumference [tex]\(C\)[/tex] of a circle is given by the formula:
[tex]\[
C = 2\pi r
\][/tex]
where [tex]\(r\)[/tex] is the radius and [tex]\(\pi\)[/tex] is a mathematical constant approximately equal to 3.14159.
2. Solve for the radius:
Given the circumference [tex]\(C = 31.4\)[/tex] units, we can solve for the radius [tex]\(r\)[/tex] using the formula:
[tex]\[
r = \frac{C}{2\pi}
\][/tex]
So,
[tex]\[
r = \frac{31.4}{2\pi}
\][/tex]
After performing the calculation, we find:
[tex]\[
r \approx 4.997465213085514 \, \text{units}
\][/tex]
3. Determine the area of the circle:
The area [tex]\(A\)[/tex] of a circle is given by the formula:
[tex]\[
A = \pi r^2
\][/tex]
Substituting the value of the radius we found:
[tex]\[
A = \pi (4.997465213085514)^2
\][/tex]
After performing the calculation, we get:
[tex]\[
A \approx 78.46020384544256 \, \text{units}^2
\][/tex]
Hence, the area of the circle with a circumference of 31.4 units is approximately [tex]\( 78.46020384544256 \)[/tex] square units.
