Geometric figure which is "snugly enclosed" by another figure
Inscribed circles of various polygonsAn inscribed triangle of a circleA tetrahedron (red) inscribed in a cube (yellow) which is, in turn, inscribed in a rhombic triacontahedron (grey). (Click here for rotating model)
In geometry, an inscribed planar shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid.[1] 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 or 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 triangles or 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 circumcircle.
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-dimensional Euclidean space, but can easily be generalized to higher dimensions and other metric spaces.
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.
^Sanders, J. Edward; Zerr, G. B. M. (1908). "193". The American Mathematical Monthly. 15 (10): 189–190. doi:10.2307/2969584. JSTOR 2969584.
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...
an arbitrary set of points, not necessarily passing through them. Inscribedfigure This article includes a list of related items that share the same name...
{2}}R^{2}\approx 2.828\,R^{2}.} In terms of the apothem r (see also inscribedfigure), the area is A = 8 tan π 8 r 2 = 8 ( 2 − 1 ) r 2 ≈ 3.314 r 2 . {\displaystyle...
19615242\,a^{2}\end{aligned}}} And in terms of the apothem r (see also inscribedfigure), the area is: A = 12 tan ( π 12 ) r 2 = 12 ( 2 − 3 ) r 2 ≃ 3.2153903...
{5}}}}\simeq 7.694208843\,a^{2}} In terms of the apothem r (see also inscribedfigure), the area is: A = 10 tan ( π 10 ) r 2 = 2 r 2 5 ( 5 − 2 5 ) ≃ 3...
by being inscribed with the name of the deceased, his titles, and often with spell 472 of the Coffin Texts or the speech of the ushabti figure found in...
square can be inscribed inside any regular polygon. The only other polygon with this property is the equilateral triangle. If the inscribed circle of a...
circumference of a circle may be defined as the limit of the perimeters of inscribed regular polygons as the number of sides increases without bound. The term...
depicts 74 men of Frederick the Great's time in life-size; some, such as the figure of his brother August, are free-standing; others are depicted in high relief...
Buddhism, by Yintuoluo and inscribed by Chushi Fanqi. Circa 1350. Hanging scroll showing Hotei, by Mokuan Reien and inscribed by Liao'an Qingyu. 14th century...
rhombus. A rhombus is a tangential quadrilateral. That is, it has an inscribed circle that is tangent to all four sides. The length of the diagonals...
and an inscribed angle of a circle are subtended by the same chord and on the same side of the chord, then the central angle is twice the inscribed angle...
for the thrice yearly showing at each village in turn the laws of Minos inscribed on brass tablets. Talos had one vein, which went from his neck to his...
it is both cyclic (has a circumscribed circle) and tangential (has an inscribed circle). The common length of the sides equals the radius of the circumscribed...
represents a reclining female figure in abstract form in bronze. It is sited on a plinth. It is inscribed "The Hampstead Figure, 1964" and signed by McWilliam...
1923, vol 3, p. 52, 88) inscribed 1. Having vertices on a curve, as in inscribedfigure. 2. Tangent to some lines, as in inscribed circle. integral An integral...
transitional period sculpture, the largest substantially complete copper Buddha figure known from the time. The statue is dated to between 500 and 700 AD (but...
In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is...
different types of pentagonal tiles using three different colors, as in the figure above right. Penrose's second tiling uses quadrilaterals called the "kite"...
tetrahedra can be inscribed in a cube, and five cubes can be inscribed in a dodecahedron, ten tetrahedra in five cubes can be inscribed in a dodecahedron:...
inscribed with verses from Quran or names of Allah and of prophets. They may also carry images of holy sites or astrological symbols. The inscribed names...
In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the...
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