Let's solve the problem step-by-step.
1. Understanding the Problem:
The circumference [tex]\(C\)[/tex] of the circle is given as [tex]\(20\pi\)[/tex] cm. We need to find the area [tex]\(A\)[/tex] of the circle in square centimeters and express it in terms of [tex]\(\pi\)[/tex].
2. Using the Circumference to Find the Radius:
The formula for the circumference of a circle is given by:
[tex]\[
C = 2\pi r
\][/tex]
where [tex]\(r\)[/tex] is the radius of the circle. Given [tex]\(C = 20\pi\)[/tex], we can set up the equation:
[tex]\[
20\pi = 2\pi r
\][/tex]
To find the radius [tex]\(r\)[/tex], divide both sides of the equation by [tex]\(2\pi\)[/tex]:
[tex]\[
r = \frac{20\pi}{2\pi} = 10 \text{ cm}
\][/tex]
3. Finding the Area Using the Radius:
The formula for the area [tex]\(A\)[/tex] of a circle is:
[tex]\[
A = \pi r^2
\][/tex]
Substituting [tex]\(r = 10\)[/tex] cm into the formula, we get:
[tex]\[
A = \pi (10)^2
\][/tex]
[tex]\[
A = \pi \times 100
\][/tex]
[tex]\[
A = 100\pi \text{ square centimeters}
\][/tex]
Therefore, the area of the circle, in terms of [tex]\(\pi\)[/tex], is [tex]\(\boxed{100\pi}\)[/tex] square centimeters.
