Circumscribed

Circumscribed is an adjective used in geometry and more generally to describe a relationship in which a figure lies around another and touches it at specific points. In planar geometry, a circle circumscribed about a polygon, or the circumcircle, is a circle that passes through all the polygon’s vertices. A polygon that has a circumcircle is called cyclic. Every triangle has a circumcircle, but not every polygon does.

A related idea is a polygon circumscribed about a circle, with the circle inscribed in the polygon. In this case the circle is tangent to all the polygon’s sides, and the polygon is said to be circumscribed about the circle (the circle is the polygon’s incircle). Such polygons are sometimes referred to as tangential polygons.

The concept extends to three dimensions. A circumsphere (circumscribed sphere) of a polyhedron is a sphere that passes through all the polyhedron’s vertices. Some polyhedra admit a circumsphere (notably the regular ones), while others do not. The term circumscribed can also describe a general enclosure: a figure drawn around another so as to touch it at designated points.

Etymology and usage: circumscribed derives from Latin circum- “around” and scribere “to write,” reflecting the idea of drawing around an object. In mathematical writing, it clarifies when a shape is drawn around another with contact at specified locations, as in circumcircle or circumsphere.