To find the circumference of a circle given by the equation \(\( x^2 + y^2 = 16 \)\), let's follow these steps:
1. Identify the equation's format:
The given equation \(\( x^2 + y^2 = 16 \)\) is in the standard form of a circle equation, \(\((x - h)^2 + (y - k)^2 = r^2\)\), where \(\((h, k)\)\) is the center and \(\(r\)\) is the radius.
2. Determine the center:
Since there are no \(\(h\)\) or \(\(k\)\) terms modified in the expression (i.e., it is in the form \(\(x^2 + y^2\)\) with no shifts), the center \(\((h, k)\)\) is at \(\((0, 0)\)\).
\(\[
\text{Center} = (0, 0)
\]\)
3. Find the radius:
The given equation \(\( x^2 + y^2 = 16 \)\) implies that \(\( r^2 = 16 \)\). To find the radius \(\( r \)\), take the square root of both sides.
\(\[
r = \sqrt{16} = 4
\]\)
4. Write the circumference formula:
The circumference \(\( C \)\) of a circle is calculated using the formula \(\( C = 2\pi r \)\).
5. Substitute the radius into the formula:
Using the radius \(\( r = 4 \)\), we substitute into the circumference formula.
\(\[
C = 2\pi \times 4 = 8\pi
\]\)
Therefore, the center of the circle is \(\((0, 0)\)\), the radius is \(\(4\)\), and the circumference of the circle is \(\( 8\pi \)\).
