1. **State the problem:** We need to find the average area of all circles with radii between 3 and 6 meters. 2. **Formula for the area of a circle:** The area $A$ of a circle with radius $r$ is given by $$A = \pi r^2$$ 3. **Understanding the average area:** Since the radius varies continuously from 3 to 6, the average area is the mean value of $\pi r^2$ over the interval $[3,6]$. 4. **Set up the integral for the average:** The average value $\bar{A}$ of a function $f(r)$ over $[a,b]$ is $$\bar{A} = \frac{1}{b-a} \int_a^b f(r) \, dr$$ Here, $f(r) = \pi r^2$, $a=3$, and $b=6$. 5. **Calculate the integral:** $$\int_3^6 \pi r^2 \, dr = \pi \int_3^6 r^2 \, dr = \pi \left[ \frac{r^3}{3} \right]_3^6 = \pi \left( \frac{6^3}{3} - \frac{3^3}{3} \right) = \pi \left( \frac{216}{3} - \frac{27}{3} \right) = \pi (72 - 9) = 63\pi$$ 6. **Calculate the average area:** $$\bar{A} = \frac{1}{6-3} \times 63\pi = \frac{1}{3} \times 63\pi = 21\pi$$ 7. **Final answer:** The average area of all circles with radii between 3 and 6 meters is $$21\pi \approx 65.97$$ square meters. This means if you pick any radius between 3 and 6 meters, the average area of the circle formed is about 65.97 square meters.