1. **Problem statement:** We have 6 points on a circle and want to find the number of polygons with 3 or 4 sides that can be formed by connecting these points. 2. **Formula and explanation:** The number of polygons with exactly $k$ sides from $n$ points is the number of combinations $\binom{n}{k}$ because any $k$ points form a polygon. 3. We want polygons with at most 4 sides, so we consider triangles ($k=3$) and quadrilaterals ($k=4$). 4. Calculate the number of triangles: $$\binom{6}{3} = \frac{6!}{3!\cdot 3!} = \frac{6\times5\times4}{3\times2\times1} = 20$$ 5. Calculate the number of quadrilaterals: $$\binom{6}{4} = \frac{6!}{4!\cdot 2!} = \frac{6\times5}{2\times1} = 15$$ 6. Total polygons with at most 4 sides: $$20 + 15 = 35$$ 7. **Answer:** The number of polygons with at most 4 sides is **35**.