To find the area of a circle given its circumference, we need to follow a series of steps involving some fundamental geometric formulas. Here’s how you can solve the problem step-by-step:
1. Recall the formula for the circumference of a circle:
The circumference [tex]\( C \)[/tex] of a circle is given by:
[tex]\[
C = 2\pi r
\][/tex]
where [tex]\( r \)[/tex] is the radius of the circle and [tex]\( \pi \)[/tex] (pi) is a mathematical constant approximately equal to 3.14159.
2. Rearrange the formula to solve for the radius [tex]\( r \)[/tex]:
We can solve for the radius [tex]\( r \)[/tex] by rearranging the equation:
[tex]\[
r = \frac{C}{2\pi}
\][/tex]
3. Substitute the given circumference into the formula:
Given that [tex]\( C = 24 \)[/tex] cm:
[tex]\[
r = \frac{24}{2\pi}
\][/tex]
4. Calculate the radius:
Performing the division:
[tex]\[
r \approx 3.8197 \, \text{cm}
\][/tex]
5. Recall the formula for the area of a circle:
The area [tex]\( A \)[/tex] of a circle is given by:
[tex]\[
A = \pi r^2
\][/tex]
6. Substitute the calculated radius into the area formula:
Using the radius we found:
[tex]\[
A = \pi \times (3.8197)^2
\][/tex]
7. Calculate the area:
Squaring the radius and then multiplying by [tex]\( \pi \)[/tex]:
[tex]\[
A \approx 45.8366 \, \text{cm}^2
\][/tex]
So, the area of the circle is approximately [tex]\( 45.837 \, \text{cm}^2 \)[/tex].
