The solid angles subtended at the vertices by all five platonic solids (regular polyhedrons) have been calculated by the author Mr H.C. Rajpoot by using standard formula of solid angle. These are the standard values of solid angles for all five platonic solids i.e. regular tetrahedron, regular hexahedron (cube), regular octahedron, regular dodecahedron & regular icosahedron useful for the analysis of platonic solids.

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slide 1:

Solid angles subtended by the platonic solids regular polyhedra at their vertices
Mr Harish Chandra Rajpoot March 2015
M.M.M. University of Technology Gorakhpur-273010 UP India
Introduction: We know that all five platonic solids i.e. regular tetrahedron regular hexahedron cube
regular octahedron regular dodecahedron regular icosahedron have all their vertices identical hence the
solid angle subtended by any platonic solid at any of its identical vertices will be equal in magnitude. If we
treat all the edges meeting at any of the identical vertices of a platonic solid as the lateral edges of a right
pyramid with a regular n-gonal base then the solid angle subtended by any of five platonic solids is calculated
by using HCR’s standard formula of solid angle. According to which solid angle subtended at the vertex
apex point by a right pyramid with a regular n-gonal base an angle between any two consecutive
lateral edges meeting at the same vertex is mathematically given by the standard generalized formula as
follows
√
Thus by setting the value of no. of edges meeting at any of the identical vertices of a platonic solid the
angle between any two consecutive edges meeting at that vertex in the above formula we can easily
calculate the solid angle subtended by the given platonic solid at its vertex. Let’s assume that the eye of
observer is located at any of the identical vertices of a given platonic solid directed focused straight to
the centre of the platonic solid as shown in the figures below then by setting the corresponding values of
in the above generalized formula we can analyse all five platonic solids as follows
1. Solid angle subtended by a regular tetrahedron at any of its four identical vertices: we know that a regular
tetrahedron has 4 congruent equilateral triangular faces 6 edges 4 identical vertices. Three equilateral
triangular faces meet at each vertex hence 3 edges meet at each vertex the angle
between any two consecutive edges is
thus in this case we have
⇒
√
√
√
√
√
√
√
Hence the solid angle
subtended by a regular tetrahedron at its vertex is given as
√
2. Solid angle subtended by a regular hexahedron cube at any of its eight identical vertices: we know that a
regular hexahedron cube has 6 congruent square faces 12 edges 8 identical vertices. Three square faces
meet at each vertex hence 3 edges meet at each vertex the angle between any two consecutive edges is
thus in this case we have
Figure 1: Eye of the observer is located
at any of four identical vertices of a
regular tetrahedron in this case

slide 2:

Solid angles subtended by the platonic solids regular polyhedra at their vertices
⇒
√
√
√
√
√
√
Hence the solid angle
subtended by a regular hexahedron cube at its vertex is
given as
3. Solid angle subtended by a regular octahedron at any of its six identical vertices: we know that a regular
octahedron has 8 congruent equilateral triangular faces 12 edges 6 identical vertices.
Four equilateral triangular faces meet at each vertex hence 4 edges meet at each
vertex the angle between any two consecutive edges is
thus in this case we have
⇒
√
√
√
√
√
√
√
√
Hence the solid angle
subtended by a regular octahedron at its vertex is given as
√
4. Solid angle subtended by a regular dodecahedron at any of its twenty identical vertices: we know that a
regular dodecahedron has 12 congruent regular pentagonal faces 30 edges 20 identical vertices. Three
regular pentagonal faces meet at each vertex hence 3 edges meet at each vertex the angle between any
two consecutive edges is
thus in this case we have
Figure 2: Eye of the observer is located
at any of eight identical vertices of a
regular hexahedron cube in this case
Figure 3: Eye of the observer is located
at any of six identical vertices of a
regular octahedron in this case

slide 3:

Solid angles subtended by the platonic solids regular polyhedra at their vertices
⇒
√
√
√
√
√
√
√
√
√
√
Hence the solid angle
subtended by a regular dodecahedron at its vertex is given as
√
√
5. Solid angle subtended by a regular icosahedron at any of its twelve identical vertices: we know that a
regular icosahedron has 20 congruent equilateral triangular faces 30 edges 12
identical vertices. Five equilateral triangular faces meet at each vertex hence 5 edges
meet at each vertex the angle between any two consecutive edges is
thus in this
case we have
⇒
√
√
√
√
√
√
√
√
√
√
√
√
Hence the solid angle
subtended by a regular icosahedron at its vertex is given as
√
√
All the above results of the solid angles subtended by the platonic solids at their vertices can be tabulated as
follows
Figure 4: Eye of the observer is located at
any of twenty identical vertices of a
regular dodecahedron in this case
Figure 5: Eye of the observer is located
at any of twelve identical vertices of a
regular icosahedron in this case

slide 4:

Solid angles subtended by the platonic solids regular polyhedra at their vertices
Platonic solid
regular
polyhedron
Regular
polygonal face
No. of
congruent
faces
No. of equal
edges
No. of
identical
vertices
Solid angle subtended by the platonic solid at
its each vertex
in Ste-radian sr
Regular tetrahedron
Equilateral
triangle
4
6
4
√
Regular hexahedron
cube
Square
6
12
8
Regular octahedron
Equilateral
triangle
8
12
6
√
Regular
dodecahedron
Regular
pentagon
12
30
20
√
√
Regular icosahedron
Equilateral
triangle
20
30
12
√
√
Note: Above articles had been derived illustrated by Mr H.C. Rajpoot B Tech Mechanical Engineering
M.M.M. University of Technology Gorakhpur-273010 UP India March 2015
Email: rajpootharishchandragmail.com
Author’s Home Page: https://notionpress.com/author/HarishChandraRajpoot

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