All Topics  
Inscribed sphere

 

 
   Email Print
   Bookmark   Link
 

 

Inscribed sphere
 
 
  Topics Inscribed sphere Topic Home

In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, the inscribed sphere or insphere of a convex
Convex

The word convex means curving out or bulging outward.Convex or convexity may refer to:Mathematics:* Convex set, a set of points containing all line segments between each pair of its points...
 polyhedron
Polyhedron

|}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
 is a sphere
Sphere

A sphere is a symmetrical geometrical object. In non-mathematical usage, the term is used to refer either to a round ball or to its two-dimensional surface....
 that is contained within the polyhedron and tangent
Tangent

In geometry, the tangent line to a curve at a given Point is the straight line that "just touches" the curve at that point . As it passes through the point of tangency, the tangent line is "going in the same direction" as the curve, and in this sense it is the best straight-line approximation to the curve at that point....
 to each of the polyhedron's faces. It is the largest sphere that is contained wholly within the polyhedron, and is dual
Dual

Dual may refer to:*a pair or a grouping of two:** dual basis, in mathematics, a basis that uniquely has a zero or unity inner product with a given basis...
 to the dual polyhedron
Dual polyhedron

In geometry, polyhedron are associated into pairs called duals, where the wikt:vertex of one correspond to the face s of the other. The dual of the dual is the original polyhedron....
's circumsphere.

All regular polyhedra have inscribed spheres, but some irregular polyhedra do not have all facets tangent to a common sphere, although it is still possible to define the largest contained sphere for such shapes.






Discussion
Ask a question about 'Inscribed sphere'
Start a new discussion about 'Inscribed sphere'
Answer questions from other users
Full Discussion Forum



Encyclopedia


In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, the inscribed sphere or insphere of a convex
Convex

The word convex means curving out or bulging outward.Convex or convexity may refer to:Mathematics:* Convex set, a set of points containing all line segments between each pair of its points...
 polyhedron
Polyhedron

|}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
 is a sphere
Sphere

A sphere is a symmetrical geometrical object. In non-mathematical usage, the term is used to refer either to a round ball or to its two-dimensional surface....
 that is contained within the polyhedron and tangent
Tangent

In geometry, the tangent line to a curve at a given Point is the straight line that "just touches" the curve at that point . As it passes through the point of tangency, the tangent line is "going in the same direction" as the curve, and in this sense it is the best straight-line approximation to the curve at that point....
 to each of the polyhedron's faces. It is the largest sphere that is contained wholly within the polyhedron, and is dual
Dual

Dual may refer to:*a pair or a grouping of two:** dual basis, in mathematics, a basis that uniquely has a zero or unity inner product with a given basis...
 to the dual polyhedron
Dual polyhedron

In geometry, polyhedron are associated into pairs called duals, where the wikt:vertex of one correspond to the face s of the other. The dual of the dual is the original polyhedron....
's circumsphere.

All regular polyhedra have inscribed spheres, but some irregular polyhedra do not have all facets tangent to a common sphere, although it is still possible to define the largest contained sphere for such shapes. For such cases, the notion of an insphere does not seem to have been properly defined and various interpretations of an insphere are to be found:
  • The sphere tangent to all faces (if one exists).
  • The sphere tangent to all face planes (if one exists).
  • The sphere tangent to a given set of faces (if one exists).
  • The largest sphere that can fit inside the polyhedron.


Often these spheres coincide, leading to confusion as to exactly what properties define the insphere for polyhedra where they do not coincide.

For example the regular small stellated dodecahedron
Small stellated dodecahedron

In geometry, the small stellated dodecahedron is a Kepler-Poinsot polyhedra. It is one of four nonconvex regular polyhedra. It is composed of 12 pentagrammic faces, with five pentagrams meeting at each vertex....
 has a sphere tangent to all faces, while a larger sphere can still be fitted inside the polyhedron. Which is the insphere? Important authorities such as Coxeter or Cundy & Rollett are clear enough that the face-tangent sphere is the insphere. Again, such authorities agree that the Archimedean polyhedra
Archimedean solid

In geometry an Archimedean solid is a highly symmetric, semi-regular convex set polyhedron composed of two or more types of regular polygons meeting in identical vertex ....
 (having regular faces and equivalent vertices) have no inspheres while the Archimedean dual or Catalan
Catalan solid

In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. The Catalan solids are named for the Belgium mathematician, Eug?ne Catalan, who first described them in 1865....
 polyhedra do have inspheres. But many authors fail to respect such distinctions and assume other definitions for the 'inspheres' of their polyhedra.

The radius of the sphere inscribed in a polyhedron P is called the inradius of P.

See also


  • Circumscribed sphere
    Circumscribed sphere

    In geometry, a circumscribed sphere of a polyhedron is a sphere that contains the polyhedron and touches each of the polyhedron's vertices. The word circumsphere is sometimes used to mean the same thing....
  • Midsphere
    Midsphere

    In geometry, the midsphere or intersphere of a polyhedron is a sphere which is tangent to every edge of the polyhedron. That is to say, it touches any given edge at exactly one point....
  • Inscribed circle


External links