1. **Problem Statement:** A circle is drawn inside a regular hexagon such that no part of the circle lies outside the hexagon. The circle does not touch all six sides. We need to find the maximum number of sides the circle can touch. 2. **Key Concept:** In a regular hexagon, the inscribed circle (incircle) touches all six sides. If a circle inside the hexagon does not touch all six sides, it must be smaller or positioned differently. 3. **Important Rule:** A circle tangent to a polygon side means it touches that side at exactly one point without crossing it. 4. **Analysis:** - The regular hexagon has six equal sides and angles. - The incircle touches all six sides. - If the circle is smaller or shifted, it can touch fewer sides. - The question is: what is the maximum number of sides it can touch without touching all six? 5. **Geometric Reasoning:** - The circle can be tangent to consecutive sides or non-consecutive sides. - Tangency to many sides requires the circle to be large and well-centered. - If the circle does not touch all six sides, it cannot be the incircle. - The maximum number of sides it can touch without touching all six is 4. 6. **Conclusion:** The maximum number of sides the circle can touch without touching all six sides is 4. **Final answer: (D) 4**