Dodecahedron

‘Dodeca’ means 12, and ‘hedra’ means face or seat. So ’dodecahedron’ is a three-dimensional solid with 12 flat faces. It is one of the 5 platonic solids. The diagram shows the shape of a dodecahedron.

By the term ‘Dodecahedron’ we generally mean a regular dodecahedron having all faces congruent.

A net of a dodecahedron can illustrate its shape from a 2-dimensional view. This net can be folded along to make a dodecahedron as shown in the video.

Parts

A dodecahedron has:

1. 12 faces (each face is a regular pentagon with 5 sides)
2. 30 edges
3. 20 vertices

Angles

The angle between 2 sides of a pentagonal face is 108°. The sum of the angles at every vertex is 3 × 108° = 324°.

2 adjacent faces intersect at 116.56505° (dihedral angle).

Formulas

Volume

Solution:

As we know,
${Vobume\left( V\right) =\dfrac{15+7\sqrt{5}}{4}e^{3}}$, here e = 5 cm
${\therefore V =\dfrac{15+7\sqrt{5}}{4}\times 5^{3}}$
= 957.89 cm3

Surface Area

Calculate the surface area of a dodecahedron whose edge length is 7 in.

Solution:

As we know,
${SurfaceArea\left( SA\right) =3\sqrt{25+10\sqrt{5}}e^{2}}$, here e = 7 in
${\therefore SA =3\sqrt{25+10\sqrt{5}}\times 7^{2}}$
= 1011.64 cm2

Real-Life Examples

1. Rubik’s dodecahedrons
2. Dodecahedron-shaped speakers
3. Dodecahedron-shaped lamps