Find the Area Difference: Circle with Radius r+2 vs Circle with Radius r

To solve this problem, we need to find the difference in areas between Circle O2 and Circle O1.

Step 1: Identify the given information
Circle O1 has a radius of $r$
Circle O2 has a radius of $r + 2$

Step 2: Recall the formula for the area of a circle
The area of a circle is given by the formula $A = \pi r^2$, where $r$ is the radius.

Step 3: Calculate the area of Circle O1
$A_1 = \pi r^2$

Step 4: Calculate the area of Circle O2
$A_2 = \pi(r+2)^2$
We need to expand $(r+2)^2$:
$(r+2)^2 = r^2 + 4r + 4$
Therefore: $A_2 = \pi(r^2 + 4r + 4) = \pi r^2 + 4\pi r + 4\pi$

Step 5: Find the difference between the two areas
The difference in area is:
$A_2 - A_1 = (\pi r^2 + 4\pi r + 4\pi) - \pi r^2$
$= \pi r^2 + 4\pi r + 4\pi - \pi r^2$
$= 4\pi r + 4\pi$

Step 6: Factor the result (optional)
We can factor out $4\pi$:
$4\pi r + 4\pi = 4\pi(r + 1)$

Therefore, Circle O2 is larger than Circle O1 by an area of $4\pi r + 4\pi$ or equivalently $4\pi(r+1)$.