CK12 - Geometry

Thereare 6 faces,12 edges,and 8 vertices.Usingthe formula:

So the figureconformsto Euler’s formula.

Example 5

In a 6-facedpolyhedron,thereare 10 edges.How manyverticesdoesthe polyhedronhave?

Use Euler'sformula.

Euler’s formula

Substitutevaluesforfande

Solve

Thereare 6 verticesin the figure.

Example 6

A 3-dimensionalfigurehas 10 vertices,5 faces,and 12 edges.It is a polyhedron?How do you know?

Use Euler'sformula.

Euler’s formula

Substitutevaluesforv,f, ande

Evaluate

The equationdoesnot hold so Euler’s formuladoesnot applyto this figure.Sinceall polyhedraconformto
Euler’s formula,this figuremustnot be a polyhedron.

RegularPolyhedra

Polyhedracan be namedand classifiedin a numberof ways—byside,by angle,by base,by numberof
faces,and so on. Perhapsthe mostimportantclassificationis whetheror not a polyhedronisregularor not.
You will recallthat aregularpolygonis a polygonwhosesidesand anglesare all congruent.

A polyhedronis regularif it has the followingcharacteristics:

• All facesare the same.


• All facesare congruentregularpolygons.


• The samenumberof facesmeetat everyvertex.