Answer :
To find the area of a circle when given the circumference, we need to follow a series of steps involving the formulas for the circumference and the area of a circle.
### Step-by-Step Solution:
1. Identify the given information and relevant formulas:
- Given the circumference (C) of the circle: [tex]\( C = 16 \)[/tex] inches
- The formula for the circumference of a circle is:
[tex]\[ C = 2 \pi r \][/tex]
- The formula for the area of a circle is:
[tex]\[ A = \pi r^2 \][/tex]
2. Solve for the radius [tex]\( r \)[/tex]:
- Use the circumference formula to solve for the radius [tex]\( r \)[/tex]:
[tex]\[ 16 = 2 \pi r \][/tex]
- Divide both sides of the equation by [tex]\( 2 \pi \)[/tex]:
[tex]\[ r = \frac{16}{2 \pi} \][/tex]
- Simplify the division:
[tex]\[ r = \frac{8}{\pi} \][/tex]
3. Calculate the area [tex]\( A \)[/tex]:
- Use the radius [tex]\( r \)[/tex] in the area formula:
[tex]\[ A = \pi r^2 \][/tex]
- Substitute [tex]\( r = \frac{8}{\pi} \)[/tex] into the area formula:
[tex]\[ A = \pi \left( \frac{8}{\pi} \right)^2 \][/tex]
- Simplify the expression inside the parentheses:
[tex]\[ \left( \frac{8}{\pi} \right)^2 = \frac{64}{\pi^2} \][/tex]
- Multiply this by [tex]\( \pi \)[/tex]:
[tex]\[ A = \pi \cdot \frac{64}{\pi^2} \][/tex]
- Simplify the multiplication (the [tex]\( \pi \)[/tex] in the numerator and one [tex]\( \pi \)[/tex] in the denominator cancel out):
[tex]\[ A = \frac{64}{\pi} \][/tex]
### Final Answer:
The area of the circle, in square inches, expressed in terms of [tex]\(\pi\)[/tex], is:
[tex]\[ A = \frac{64}{\pi} \, \text{in}^2 \][/tex]
Submit this answer as [tex]\(\frac{64}{\pi} \, \text{in}^2\)[/tex].
### Step-by-Step Solution:
1. Identify the given information and relevant formulas:
- Given the circumference (C) of the circle: [tex]\( C = 16 \)[/tex] inches
- The formula for the circumference of a circle is:
[tex]\[ C = 2 \pi r \][/tex]
- The formula for the area of a circle is:
[tex]\[ A = \pi r^2 \][/tex]
2. Solve for the radius [tex]\( r \)[/tex]:
- Use the circumference formula to solve for the radius [tex]\( r \)[/tex]:
[tex]\[ 16 = 2 \pi r \][/tex]
- Divide both sides of the equation by [tex]\( 2 \pi \)[/tex]:
[tex]\[ r = \frac{16}{2 \pi} \][/tex]
- Simplify the division:
[tex]\[ r = \frac{8}{\pi} \][/tex]
3. Calculate the area [tex]\( A \)[/tex]:
- Use the radius [tex]\( r \)[/tex] in the area formula:
[tex]\[ A = \pi r^2 \][/tex]
- Substitute [tex]\( r = \frac{8}{\pi} \)[/tex] into the area formula:
[tex]\[ A = \pi \left( \frac{8}{\pi} \right)^2 \][/tex]
- Simplify the expression inside the parentheses:
[tex]\[ \left( \frac{8}{\pi} \right)^2 = \frac{64}{\pi^2} \][/tex]
- Multiply this by [tex]\( \pi \)[/tex]:
[tex]\[ A = \pi \cdot \frac{64}{\pi^2} \][/tex]
- Simplify the multiplication (the [tex]\( \pi \)[/tex] in the numerator and one [tex]\( \pi \)[/tex] in the denominator cancel out):
[tex]\[ A = \frac{64}{\pi} \][/tex]
### Final Answer:
The area of the circle, in square inches, expressed in terms of [tex]\(\pi\)[/tex], is:
[tex]\[ A = \frac{64}{\pi} \, \text{in}^2 \][/tex]
Submit this answer as [tex]\(\frac{64}{\pi} \, \text{in}^2\)[/tex].
