1. **State the problem:** We have a circular garden with radius $1.5$ m and a regular hexagonal patio inscribed inside it. We need to find the area covered by grass, which is the area inside the circle but outside the hexagon. 2. **Formulas and important rules:** - Area of a circle: $$A_{circle} = \pi r^2$$ where $r$ is the radius. - A regular hexagon can be divided into 6 equilateral triangles. - Area of a regular hexagon with side length $s$: $$A_{hexagon} = \frac{3\sqrt{3}}{2} s^2$$ - For a regular hexagon inscribed in a circle, the radius of the circle equals the side length of the hexagon: $$s = r$$ 3. **Calculate the area of the circle:** $$A_{circle} = \pi (1.5)^2 = \pi \times 2.25 = 2.25\pi$$ 4. **Calculate the side length of the hexagon:** Since the hexagon is inscribed, $$s = r = 1.5$$ m. 5. **Calculate the area of the hexagon:** $$A_{hexagon} = \frac{3\sqrt{3}}{2} (1.5)^2 = \frac{3\sqrt{3}}{2} \times 2.25 = \frac{3\sqrt{3}}{2} \times 2.25$$ Simplify: $$= 3\sqrt{3} \times 1.125 = 3.375\sqrt{3}$$ 6. **Calculate the area covered by grass:** This is the area inside the circle but outside the hexagon: $$A_{grass} = A_{circle} - A_{hexagon} = 2.25\pi - 3.375\sqrt{3}$$ 7. **Approximate the numerical value:** Using $\pi \approx 3.1416$ and $\sqrt{3} \approx 1.732$: $$A_{grass} \approx 2.25 \times 3.1416 - 3.375 \times 1.732 = 7.0686 - 5.847 = 1.2216$$ m$^2$ approximately. **Final answer:** The area covered by grass is approximately $1.22$ m$^2$.