Inscribed figure

Inscribed circles of various polygons

An inscribed triangle of a circle

In geometry, an inscribed planar shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid. To say that "Figure F is inscribed in figure G" means precisely the same thing as "figure G is circumscribed about figure F". A circle or ellipse inscribed in a convex polygon (or a sphere or ellipsoid inscribed in a convex polyhedron) is tangent to every side of the outer figure (but see Inscribed sphere for semantic variants). A polygon inscribed in a circle, ellipse, or polygon (or a polyhedron inscribed in a sphere, ellipsoid, or polyhedron) has each vertex on the outer figure; if the outer figure is a polygon or polyhedron, there must be a vertex of the inscribed polygon or polyhedron on each side of the outer figure. Familiar examples include circles inscribed in triangles or regular polygons, and triangles or regular polygons inscribed in circles.

An inscribed figure is not necessarily unique in orientation; this can easily be seen, for example, when the given outer figure is a circle, in which case a rotation of an inscribed figure gives another inscribed figure that is congruent to the original one.

The definition given above assumes that the objects concerned are embedded in two- or three-dimensional Euclidean space, but can easily be generalized to higher dimensions and other metric spaces.

The inradius or filling radius of a given outer figure is the radius of the inscribed circle or sphere, if it exists.

Facts about inscribed figures

• Every circle has an inscribed triangle with any three given angle measures (summing of course to 180°), and every triangle can be inscribed in some circle (which is called its circumscribed circle).

• Every triangle has an inscribed circle, called the incircle.

• Every circle has an inscribed regular polygon of n sides, for any n≥3, and every regular polygon can be inscribed in some circle.

• Every regular polygon has an inscribed circle, and every circle can be inscribed in some regular polygon of n sides, for any n≥3.

• Every triangle has an infinitude of inscribed ellipses. One of them is a circle, and one of them is the Steiner inellipse which is tangent to the triangle at the midpoints of the sides.

• Every triangle has three inscribed squares, though two of them coincide with each other in the case of a right triangle.

See also

• Circumconic and inconic
• Inscribed square problem