Question

the circumference of a circle is ( 16pi ) inches. which of the following is the area, in inches², of the circle?

reference sheet
formulas and diagrams:

• circle: ( a = pi r^2 ), ( c = 2pi r )
• rectangle: ( a = lw )
• triangle: ( a = \frac{1}{2}bh )
• right triangle: ( c^2 = a^2 + b^2 )
• special right triangles (30-60-90, 45-45-90)
• rectangular prism: ( v = lwh )
• cylinder: ( v = pi r^2 h )
• sphere: ( v = \frac{4}{3}pi r^3 )
• cone: ( v = \frac{1}{3}pi r^2 h )
• pyramid: ( v = \frac{1}{3}lwh )

the number of degrees of arc in a circle is 360.
the number of radians of arc in a circle is ( 2pi ).
the sum of the measures in degrees of the angles of a triangle is 180.

options:
( 8 )
( 8pi )
( 16pi )
( 64pi )

Explanation:

Step1: Recall the circumference formula

The formula for the circumference \( C \) of a circle is \( C = 2\pi r \), where \( r \) is the radius. We know \( C = 16\pi \), so we set up the equation:
\( 16\pi = 2\pi r \)

Step2: Solve for the radius \( r \)

Divide both sides of the equation by \( 2\pi \):
\( r=\frac{16\pi}{2\pi}=8 \)

Step3: Recall the area formula

The formula for the area \( A \) of a circle is \( A=\pi r^{2} \).

Step4: Substitute \( r = 8 \) into the area formula

\( A=\pi(8)^{2}=\pi\times64 = 64\pi \)? Wait, no, wait. Wait, no, wait, let's check again. Wait, \( r = 8 \)? Wait, \( 16\pi=2\pi r \), so dividing both sides by \( 2\pi \), \( r = 8 \)? Wait, no, \( 16\pi\div2\pi = 8 \), so \( r = 8 \)? Wait, but then area is \( \pi r^{2}=\pi\times8^{2}=64\pi \)? But wait, the options have 64π? Wait, no, wait, maybe I made a mistake. Wait, no, let's check again. Wait, circumference is \( 2\pi r = 16\pi \), so \( r = 8 \). Then area is \( \pi r^{2}=\pi\times8^{2}=64\pi \). But wait, the options: 8, 8π, 16π, 64π. So 64π is an option. Wait, but let me check again. Wait, maybe I messed up the radius. Wait, \( C = 2\pi r \), so \( r=\frac{C}{2\pi}=\frac{16\pi}{2\pi}=8 \). Then area \( A=\pi r^{2}=\pi\times8^{2}=64\pi \). So the answer should be 64π? Wait, but let me check the options again. The options are 8, 8π, 16π, 64π. So 64π is the last option. Wait, but maybe I made a mistake. Wait, no, let's do it again. Circumference \( C = 2\pi r = 16\pi \). Divide both sides by \( 2\pi \): \( r = 8 \). Then area \( A=\pi r^{2}=\pi\times8^{2}=64\pi \). Yes, that's correct.

Wait, but wait, maybe the question was misread. Wait, the circumference is 16π, so radius is 8, area is 64π. So the answer is 64π.

Answer:

\( 64\pi \) (corresponding to the option with 64π)