To find the area of a circle when given the circumference, we can follow these steps:
1. Understand the relationship between circumference and radius: The circumference (C) of a circle is related to its radius (r) by the formula:
[tex]\[ C = 2 \pi r \][/tex]
where [tex]\(\pi\)[/tex] (pi) is a mathematical constant approximately equal to 3.14159.
2. Solve for the radius: Given that the circumference is [tex]\(19\pi\)[/tex] cm, we can use the formula to solve for the radius:
[tex]\[ 19\pi = 2\pi r \][/tex]
Dividing both sides by [tex]\(2\pi\)[/tex] gives us:
[tex]\[ r = \frac{19\pi}{2\pi} \][/tex]
Simplifying that, we find that the radius [tex]\(r\)[/tex] is:
[tex]\[ r = \frac{19}{2} \][/tex]
[tex]\[ r = 9.5 \text{ cm} \][/tex]
3. Calculate the area: Now that we have the radius, we can find the area (A) of the circle using the formula:
[tex]\[ A = \pi r^2 \][/tex]
Plugging the radius into the formula:
[tex]\[ A = \pi \times (9.5)^2 \][/tex]
[tex]\[ A = \pi \times 90.25 \][/tex]
[tex]\[ A = 90.25\pi \text{ cm}^2 \][/tex]
So, the area of the circle is [tex]\(90.25\pi\)[/tex] square centimeters when expressed in terms of [tex]\(\pi\)[/tex].
