Solid angles subtended by the platonic solids at their vertices by HCR

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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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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

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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

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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

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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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