geometry, an inscribedplanarshape 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
ellipse inscribed in a
convex polygon (or a
ellipsoid inscribed in a
convex polyhedron) is
tangent to every
face 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. 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.
Familiar examples of inscribed figures include circles inscribed in
regular polygons, and triangles or regular polygons inscribed in circles. A circle inscribed in any polygon is called its
incircle, in which case the polygon is said to be a
tangential polygon. A polygon inscribed in a circle is said to be a
cyclic polygon, and the circle is said to be its circumscribed circle or
The inradius or
filling radius of a given outer figure is the
radius of the inscribed circle or sphere, if it exists.
The definition given above assumes that the objects concerned are embedded in two- or three-
dimensionalEuclidean space, but can easily be generalized to higher dimensions and other
For an alternative usage of the term "inscribed", see the
inscribed square problem, in which a square is considered to be inscribed in another figure (even a non-convex one) if all four of its vertices are on that figure.
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 or circumcircle).
Every triangle has an inscribed circle, called the
Every circle has an inscribed regular polygon of n sides, for any n≥3, and every regular polygon can be inscribed in some circle (called its circumcircle).
Every regular polygon has an inscribed circle (called its incircle), and every circle can be inscribed in some regular polygon of n sides, for any n≥3.
Not every polygon with more than three sides has an inscribed circle; those polygons that do are called
tangential polygons. Not every polygon with more than three sides is an inscribed polygon of a circle; those polygons that are so inscribed are called
Every triangle can be inscribed in an ellipse, called its
Steiner circumellipse or simply its Steiner ellipse, whose center is the triangle's
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 acute triangle has
three inscribed squares. In a right triangle two of them are merged and coincide with each other, so there are only two distinct inscribed squares. An obtuse triangle has a single inscribed square, with one side coinciding with part of the triangle's longest side.