1. **State the problem:** We need to find the two possible values of $r$ that satisfy the equation $$\frac{216}{r^2} + 19 = 25.$$\n\n2. **Isolate the fraction:** Subtract 19 from both sides to isolate the fraction term:\n$$\frac{216}{r^2} = 25 - 19$$\n$$\frac{216}{r^2} = 6.$$\n\n3. **Solve for $r^2$:** Multiply both sides by $r^2$ and then divide both sides by 6 to solve for $r^2$:\n$$\cancel{\frac{216}{r^2}} \times r^2 = 6 \times r^2$$\n$$216 = 6r^2$$\n$$\frac{\cancel{216}}{6} = \frac{6r^2}{6}$$\n$$36 = r^2.$$\n\n4. **Find $r$:** Take the square root of both sides to find $r$:\n$$r = \pm \sqrt{36}$$\n$$r = \pm 6.$$\n\n**Final answer:** The two possible values of $r$ are $6$ and $-6$.