To find the radius of a circle where the area and circumference are equal, we can start with the formulas for area and circumference:
- Area (A) = πr²
- Circumference (C) = 2πr
We want to set these two equations equal to each other:
A = C
πr² = 2πr
Next, we can simplify by dividing both sides by π (assuming π ≠ 0):
r² = 2r
Now, we can rearrange the equation:
r² – 2r = 0
Factoring this equation gives us:
r(r – 2) = 0
Setting each factor to zero, we get:
- r = 0
- r = 2
Since a radius cannot be zero in the context of a circle, we disregard r = 0.
Thus, the only valid solution is:
r = 2
This means that the radius of the circle where its area and circumference have the same value is 2 units. To verify:
When r = 2:
- Area: A = π(2)² = 4π
- Circumference: C = 2π(2) = 4π
Indeed, when the radius is 2, the area and circumference both equal 4π, confirming that our solution is correct!