1. **State the problem:** Find the area of the shaded region, which is the area of the sector of a circle with radius $32$ and central angle $45^\circ$ minus the area of the triangle formed by the two radii and the chord. 2. **Formula for sector area:** The area of a sector with radius $r$ and central angle $\theta$ (in degrees) is given by $$\text{Area}_{\text{sector}} = \frac{\theta}{360} \pi r^2$$ 3. **Calculate the sector area:** $$\text{Area}_{\text{sector}} = \frac{45}{360} \pi (32)^2 = \frac{1}{8} \pi \times 1024 = 128\pi$$ 4. **Find the area of the triangle:** The triangle is isosceles with two sides equal to the radius $32$ and an included angle of $45^\circ$. The area of a triangle with two sides $a$, $b$ and included angle $\theta$ is $$\text{Area}_{\triangle} = \frac{1}{2}ab \sin(\theta)$$ Here, $a = b = 32$, $\theta = 45^\circ$. 5. **Calculate the triangle area:** $$\text{Area}_{\triangle} = \frac{1}{2} \times 32 \times 32 \times \sin(45^\circ) = 512 \times \frac{\sqrt{2}}{2} = 256\sqrt{2}$$ 6. **Approximate the triangle area:** $$256\sqrt{2} \approx 256 \times 1.4142 = 362.0$$ 7. **Find the shaded area:** $$\text{Shaded area} = \text{Area}_{\text{sector}} - \text{Area}_{\triangle} = 128\pi - 256\sqrt{2}$$ 8. **Approximate the shaded area:** $$128\pi \approx 128 \times 3.1416 = 402.1$$ $$\text{Shaded area} \approx 402.1 - 362.0 = 40.1$$ 9. **Final answer:** The shaded area is $$128\pi - 256\sqrt{2} \approx 40.1$$ This matches none of the options exactly, but the problem's options combine $\pi$ terms and decimals. The closest matching form is $128\pi + 45.2$ if the decimal is the difference, but the correct exact form is $128\pi - 256\sqrt{2}$. Since the problem asks to leave the answer in terms of $\pi$ and round decimals, the shaded area is: $$128\pi - 362.0$$ or approximately $40.1$. --- "slug": "sector shaded", "subject": "geometry", "desmos": {"latex": "y=0", "features": {"intercepts": true, "extrema": true}}, "q_count": 1