Piramide pentagonale elongata

Piramide pentagonale elongata

Tipo	Solido di Johnson J₈ - J₉ - J₁₀
Forma facce	5 Triangoli 5 Quadrati 1 Pentagono
Nº facce	11
Nº spigoli	20
Nº vertici	11
Incidenza dei vertici	5(4².5) 5(3².4²) 1(3⁵)
Gruppo di simmetria	C_5v, [5], (*55)
Gruppo rotazionale	C₅, [5]⁺, (55)
Duale	Autoduale
Proprietà	Convessità
Politopi correlati
Poliedro duale
Sviluppo piano

Manuale

In geometria solida, la piramide pentagonale elongata è un solido con 11 facce che può essere costruito, come intuibile dal suo nome, allungando una piramide pentagonale attraverso l'aggiunta di un prisma pentagonale alla sua base.

Caratteristiche

Nel caso in cui tutte le sue facce siano poligoni regolari, la piramide pentagonale elongata diventa uno dei 92 solidi di Johnson, in particolare quello indicato come J₉, ossia un poliedro strettamente convesso avente come facce dei poligoni regolari ma comunque non appartenente alla famiglia dei poliedri uniformi.^[1]

Come ogni altra piramide elongata, anche questa è un solido topologicamente, ma non geometricamente, autoduale.^[2]

Formule

Considerando una piramide pentagonale elongata avente come facce dei poligoni regolari aventi lato di lunghezza $a$ , le formule per il calcolo del volume $V$ , della superficie $A$ e dell'altezza $h$ risultano essere:

V={\frac {a^{3}}{24}}\left(5+{\sqrt {5}}+6{\sqrt {25+10{\sqrt {5}}}}\right)\approx 2,021980233\cdot a^{3};

A={\frac {a^{2}}{4}}\left(20+5{\sqrt {3}}+{\sqrt {25+10{\sqrt {5}}}}\right)\approx 8,88554091\cdot a^{2};

h=a\left(1+{\sqrt {\frac {5-{\sqrt {5}}}{10}}}\right)\approx 1,525731112\cdot a.

Poliedro duale

Topologicamente la piramide pentagonale elongata è autoduale. Geometricamente, il suo poliedro duale ha facce diverse: un pentagono, cinque triangoli isosceli e cinque trapezi isosceli.

Poliedro duale	Sviluppo piano del duale

Note

^ Norman W. Johnson, Convex Polyhedra with Regular Faces, in Canadian Journal of Mathematics, vol. 18, Canadian Mathematical Society, 1966, pp. 169-200, DOI:10.4153/CJM-1966-021-8. URL consultato il 14 luglio 2021.
^ (EN) Eric W. Weisstein, Piramide pentagonale elongata, in MathWorld, Wolfram Research. URL consultato il 9 luglio 2021.

Collegamenti esterni

(EN) Eric W. Weisstein, Piramide pentagonale elongata, su MathWorld, Wolfram Research.

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