To find the radius of a circle where the area and circumference are equal, we start with the formulas for area and circumference:
- Area (A) = πr²
- Circumference (C) = 2πr
We want to set the area equal to the circumference:
A = C
Thus, we can write:
πr² = 2πr
We can simplify this equation by dividing both sides by π (assuming π is not zero):
r² = 2r
Next, we can rearrange the equation:
r² - 2r = 0
Now, we can factor the left side:
r(r - 2) = 0
Setting each factor to zero gives us:
- r = 0
- r – 2 = 0 → r = 2
The radius cannot be zero, as it would not form a circle. Therefore, the only valid solution is:
r = 2
This means the radius of the circle where the area and circumference are equal is 2 units.