This document discusses calculating the volume of a dodecahedron, which is a polyhedron with 12 pentagonal faces. It provides the formulas to calculate the volume of a single pentagonal pyramid and the total volume of the dodecahedron. An example is worked through to calculate the area and total volume of a dodecahedron where the base triangles have an area of 900 cm^2, the apotema is 3 cm, the height is 4 cm, and the edge length is 10 cm. The total volume calculated is 8,863.1189 cm^3.

1. ELS
POLiedres
estrellats

2. DODECÀEDRE
=
12 CARES
PENTAGONALS
=
12 PIRÀMIDES DE
BASE PENTAGONAL

3. 12 PIRÀMIDES= 36
TRIANGLES
ÀREA 36 TRIANGLES=
B·H/2 · 36
VOLUM= VOLUM 12 PIRÀMIDES +
VOLUM DODECÀEDRE
VOLUM PIRÀMIDE
PENTAGONAL
=
ÀREA DE LA
BASE·ALTURA DE LA
PIRÀMIDE /3
VOLUM DODECÀEDRE
7,6631189 · A^3

4. EXEMPLE
10 cm
A (triangles)= B · H/2 ·36
A= 10 · 5/2 ·36
A= 50/2 · 36
A= 25 · 36
A= 900 cm^2
5 cm

5. VOLUM
VOLUM PIRÀMIDE
PENTAGONAL
=
P·a/2 · h/3
=
P·a·h/6
DONEM COM A
DADES:
APOTEMA= 3cm
ALTURA= 4cm
ARESTA= 10cm
VOLUM 12
PIRÀMIDES=
50·3·4/6 ·12=
1200cm^3
VOLUM
DODECÀEDRE=
7,6631189 · 10^3=
7663,1189 cm^3
VOLUM
TOTAL=
1.200+
7.663,1189
=
8.863,1189
Cm^3

6. RESULTAT
ÀREA DODECÀEDRE ESTELAT = 900 cm^2
VOLUM DODECÀEDRE ESTELAT = 8.863,1189 cm^3

7. POLIEDRES A LA VIDA REAL