Inscribed and Circumscribed Polygons

When a regular polygon sits inscribed in a circle, every vertex touches the circle, so every corner is a distance r (the radius) from the center. Draw a segment from the center to each vertex, and the polygon splits into congruent isosceles triangles, each with two sides of length r and a vertex angle of 360/n, where n is the number of sides. That's the whole mechanism: any question about inscribed polygon side lengths is really a question about one of these triangles.

A square inscribed in a circle is the cleanest case, because its diagonal passes through the center and equals the diameter exactly. That diagonal cuts the square into two 45-45-90 triangles, which is why the side length comes out to diameter over √2. For a hexagon, the central triangles are equilateral (since 360/6 = 60 degrees and the triangle is isosceles), which is why the hexagon's side length equals the radius itself — a shortcut worth memorizing.

The key misconception to clear up: an inscribed polygon's perimeter is always less than the circle's circumference, because straight chords are shorter than the arcs they cut off. As n grows, the polygon hugs the circle more closely and its perimeter climbs toward the circumference — but never reaches it, no matter how many sides you add. This idea connects directly to arc length and sector area questions, since both inscribed polygons and arcs are built from the same central-angle triangles.