1. **State the problem:** We have a circle with center $T$ and chord $UV$ of length $4\pi$. The central angle $\angle UTV$ is $90^\circ$. We need to find the area of the shaded region, which is the part of the circle outside the $90^\circ$ sector, i.e., the remaining $270^\circ$ sector. 2. **Identify known values and formulas:** - Length of chord $UV = 4\pi$ - Central angle $\angle UTV = 90^\circ$ - The shaded area corresponds to $270^\circ$ of the circle. 3. **Find the radius $r$ of the circle:** The chord length formula for a chord subtending an angle $\theta$ at the center is: $$UV = 2r \sin\left(\frac{\theta}{2}\right)$$ Here, $\theta = 90^\circ$, so: $$4\pi = 2r \sin\left(\frac{90^\circ}{2}\right) = 2r \sin(45^\circ)$$ Since $\sin(45^\circ) = \frac{\sqrt{2}}{2}$, we have: $$4\pi = 2r \times \frac{\sqrt{2}}{2} = r \sqrt{2}$$ 4. **Solve for $r$:** $$r = \frac{4\pi}{\sqrt{2}} = 4\pi \times \frac{\sqrt{2}}{2} = 2\pi \sqrt{2}$$ 5. **Calculate the total area of the circle:** $$\text{Area} = \pi r^2 = \pi \left(2\pi \sqrt{2}\right)^2 = \pi \times 4\pi^2 \times 2 = 8\pi^3$$ 6. **Calculate the area of the $90^\circ$ sector:** The area of a sector is: $$\text{Sector area} = \frac{\theta}{360^\circ} \times \pi r^2$$ For $\theta = 90^\circ$: $$\text{Sector area} = \frac{90}{360} \times 8\pi^3 = \frac{1}{4} \times 8\pi^3 = 2\pi^3$$ 7. **Calculate the shaded area (the remaining $270^\circ$ sector):** $$\text{Shaded area} = \text{Total area} - \text{Sector area} = 8\pi^3 - 2\pi^3 = 6\pi^3$$ 8. **Express the answer as a fraction times $\pi$:** $$6\pi^3 = 6\pi \times \pi^2$$ Since the problem asks for the answer as a fraction times $\pi$, we keep it as: $$6\pi^3$$ **Final answer:** $$\boxed{6\pi^3}$$