Here's how you can solve the problem step-by-step. We are given that the area of the circle is two times its circumference and need to find the diameter of the circle. We'll use the formulas for the area and circumference of a circle and solve for the radius, then calculate the diameter.
1. Recall the formulas for the area and circumference of a circle:
- Area [tex]\( A \)[/tex] of a circle is given by [tex]\( A = \pi r^2 \)[/tex]
- Circumference [tex]\( C \)[/tex] of a circle is given by [tex]\( C = 2 \pi r \)[/tex]
2. Given condition: The area is two times the circumference.
[tex]\[
\pi r^2 = 2 \times (2 \pi r)
\][/tex]
3. Simplify the equation:
[tex]\[
\pi r^2 = 4 \pi r
\][/tex]
4. Divide both sides by [tex]\( \pi \)[/tex] (assuming [tex]\( \pi \neq 0 \)[/tex]):
[tex]\[
r^2 = 4r
\][/tex]
5. Rearrange the equation to form a standard quadratic equation:
[tex]\[
r^2 - 4r = 0
\][/tex]
6. Factorize the quadratic equation:
[tex]\[
r(r - 4) = 0
\][/tex]
7. Solve for [tex]\( r \)[/tex]:
[tex]\[
r = 0 \quad \text{or} \quad r = 4
\][/tex]
Since the radius cannot be zero, we discard [tex]\( r = 0 \)[/tex].
Therefore, [tex]\( r = 4 \)[/tex] cm.
8. Calculate the diameter of the circle:
The diameter [tex]\( d \)[/tex] is twice the radius.
[tex]\[
d = 2r = 2 \times 4 = 8 \, \text{cm}
\][/tex]
Thus, the diameter of the circle is [tex]\( 8 \)[/tex] cm.
