'Inscribed' means *snugly inside*; 'circumscribed' means *drawn around the outside*. Put a polygon and a circle together and either one can play either role.
Build a polygon
regular hexagon
interior each
120.0°
120.0°
exterior each
60.0°
60.0°
interior sum
720°
720°
diagonals
9
9
Two pairings
- Polygon inscribed in a circle — every vertex of the polygon sits on the circle. The circle is then *circumscribed* about the polygon; its centre is the circumcentre and its radius the circumradius.
- Circle inscribed in a polygon — the circle touches every side from the inside. It's the *incircle*; its centre is the incentre and its radius the inradius.
- A polygon that has a circumscribed circle is called cyclic. Every triangle is cyclic; so is every regular polygon.
- For a regular n-gon both circles share the same centre — the polygon is sandwiched between its incircle and circumcircle.
Your turn
A square is inscribed in a circle of radius 5. How long is the square's diagonal?
Crank the slider above up to 20-something sides — the regular polygon hugs its circle tighter and tighter. That squeeze (polygon between incircle and circumcircle) is exactly how Archimedes first pinned down π.