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Full-Text Articles in Education

Derivation Of The Volume Of Tetrahedron/Pyramid Bounded By A Given Plane & The Co-Ordinate Planes, Harish Chandra Rajpoot Rajpoot Hcr Nov 2015

Derivation Of The Volume Of Tetrahedron/Pyramid Bounded By A Given Plane & The Co-Ordinate Planes, Harish Chandra Rajpoot Rajpoot Hcr

Harish Chandra Rajpoot H.C. Rajpoot

The article here deals with the derivation of a general expression to calculate the volume of tetrahedron/pyramid bounded by a given plane & the co-rdinate planes (i.e. XY-plane, YZ-plane & ZX-plane) using intercept form of equation of a plane in 3-D space. All the derivations are based on simple geometry. These are very useful to directly calculate the volume of the bounded tetrahedron/pyramid.


Mathematical Analysis Of Sphere Resting In The Vertex Of Right Pyramid & Polyhedron, Filleting Of The Faces & Packing Of The Spheres In The Vertex, Harish Chandra Rajpoot Rajpoot Hcr Jun 2015

Mathematical Analysis Of Sphere Resting In The Vertex Of Right Pyramid & Polyhedron, Filleting Of The Faces & Packing Of The Spheres In The Vertex, Harish Chandra Rajpoot Rajpoot Hcr

Harish Chandra Rajpoot H.C. Rajpoot

The generalized formula derived here by the author are applicable to locate any sphere, with a certain radius, resting in a vertex (corner) at which n no. of edges meet together at angle α between any two consecutive of them such as the vertex of platonic solids, any of two identical & diagonally opposite vertices of uniform polyhedrons with congruent right kite faces & the vertex of right pyramid with regular n-gonal base. These are also useful for filleting the faces meeting at the vertex of the polyhedron to best fit the sphere in that vertex. These are used to determine the ...


Identical Circles Touching One Another On The Spherical Polyhedrons Analogous To Archimedean Solids, Harish Chandra Rajpoot Rajpoot Hcr May 2015

Identical Circles Touching One Another On The Spherical Polyhedrons Analogous To Archimedean Solids, Harish Chandra Rajpoot Rajpoot Hcr

Harish Chandra Rajpoot H.C. Rajpoot

The formula, derived here by the author H.C. Rajpoot, are applicable on a certain no. of the identical circles touching one another at different points, centered at the identical vertices of a spherical polyhedron analogous to an Archimedean solid for calculating the different parameters such as flat radius & arc radius of each circle, total surface area covered by all the circles, percentage of surface area covered etc. These formula are very useful for tiling, packing the identical circles in different patterns & analyzing the spherical surfaces analogous to all 13 Archimedean solids. Thus also useful in designing & modelling of tiled ...


Identical Circles Touching One Another On A Whole (Entire) Spherical Surface, Harish Chandra Rajpoot Rajpoot Hcr May 2015

Identical Circles Touching One Another On A Whole (Entire) Spherical Surface, Harish Chandra Rajpoot Rajpoot Hcr

Harish Chandra Rajpoot H.C. Rajpoot

All the articles discussed & analysed here are related to all five platonic solids. A certain no. of the identical circles are touching one another on a whole (entire) spherical surface having certain radius then all the important parameters such as flat radius & arc radius of each circle, total surface area & its percentage covered by all the circles on the sphere have been easily calculated by using simple geometry & table for the important parameters of all five platonic solids by the author Mr H.C. Rajpoot. These parameters are very useful for drawing the identical circles on a spherical surface & for ...


Reflection Of A Point About A Line & A Plane In 2-D & 3-D Co-Ordinate Systems, Harish Chandra Rajpoot Rajpoot Hcr May 2015

Reflection Of A Point About A Line & A Plane In 2-D & 3-D Co-Ordinate Systems, Harish Chandra Rajpoot Rajpoot Hcr

Harish Chandra Rajpoot H.C. Rajpoot

All the articles are related to the reflection of any point about a line in 2-D co-ordinate system and about a line & a plane in 3-D co-ordinate system. Point of reflection about a line or a plane can be easily determined simply by applying the procedures explained or by using formula derived here. These formulas are also useful to determine the foot of perpendicular drawn from a point to a line or a plane in 3-D space. All these derivations are based on the application of simple geometry.


Solid Angles Subtended By The Platonic Solids (Regular Polyhedra) At Their Vertices, Harish Chandra Rajpoot Rajpoot Hcr Mar 2015

Solid Angles Subtended By The Platonic Solids (Regular Polyhedra) At Their Vertices, Harish Chandra Rajpoot Rajpoot Hcr

Harish Chandra Rajpoot H.C. Rajpoot

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.


Mathematical Analysis Of Tetrahedron (Solid Angle Subtended By Any Tetrahedron At Its Vertex), Harish Chandra Rajpoot Rajpoot Hcr Mar 2015

Mathematical Analysis Of Tetrahedron (Solid Angle Subtended By Any Tetrahedron At Its Vertex), Harish Chandra Rajpoot Rajpoot Hcr

Harish Chandra Rajpoot H.C. Rajpoot

All the articles have been derived by the author Mr H.C. Rajpoot by using HCR's Inverse cosine formula & HCR's Theory of Polygon. These formula are very practical & simple to apply in case of any tetrahedron to calculate the internal (dihedral) angles between the consecutive lateral faces meeting at any of four vertices & the solid angle subtended by it (tetrahedron) at the vertex when the angles between the consecutive edges meeting at the same vertex are known. These are the generalized formula which can also be applied in case of three faces meeting at the vertex of various ...


Mathematical Analysis Of Great Rhombicuboctahedron (An Archimedean Solid) By H.C. Rajpoot, Harish Chandra Rajpoot Rajpoot Hcr Mar 2015

Mathematical Analysis Of Great Rhombicuboctahedron (An Archimedean Solid) By H.C. Rajpoot, Harish Chandra Rajpoot Rajpoot Hcr

Harish Chandra Rajpoot H.C. Rajpoot

All the important parameters of a great rhombicuboctahedron (an Archimedean solid), having 12 congruent square faces, 8 regular hexagonal faces, 6 congruent regular octagonal faces each of equal edge length, 72 edges & 48 vertices lying on a spherical surface with certain radius, have been derived by the author H.C. Rajpoot by applying "HCR's Theory of Polygon" to calculate the solid angle subtended by each square face, regular hexagonal face & regular octagonal face & their normal distances from the center of great rhombicuboctahedron, dihedral angles between the adjacent faces, inscribed radius, circumscribed radius, mean radius, surface area & volume. These formula ...


Mathematical Analysis Of Great Rhombicosidodecahedron (The Largest Archimedean Solid) By H.C. Rajpoot, Harish Chandra Rajpoot Rajpoot Hcr Mar 2015

Mathematical Analysis Of Great Rhombicosidodecahedron (The Largest Archimedean Solid) By H.C. Rajpoot, Harish Chandra Rajpoot Rajpoot Hcr

Harish Chandra Rajpoot H.C. Rajpoot

All the important parameters of a great rhombicosidodecahedron (the largest Archimedean solid), having 30 congruent square faces, 20 regular hexagonal faces, 12 congruent regular decagonal faces each of equal edge length, 180 edges & 120 vertices lying on a spherical surface with certain radius, have been derived by the author Mr H.C. Rajpoot by applying "HCR's Theory of Polygon" to calculate the solid angle subtended by each square face, regular hexagonal face & regular decagonal face & their normal distances from the center of great rhombicosidodecahedron, dihedral angles between the adjacent faces, inscribed radius, circumscribed radius, mean radius, surface area & volume ...


Mathematical Analysis Of A Uniform Tetradecahedron With 2 Congruent Regular Hexagonal Faces, 12 Congruent Trapezoidal Faces & 18 Vertices Lying On A Spherical Surface By Hcr, Harish Chandra Rajpoot Rajpoot Hcr Mar 2015

Mathematical Analysis Of A Uniform Tetradecahedron With 2 Congruent Regular Hexagonal Faces, 12 Congruent Trapezoidal Faces & 18 Vertices Lying On A Spherical Surface By Hcr, Harish Chandra Rajpoot Rajpoot Hcr

Harish Chandra Rajpoot H.C. Rajpoot

All the important parameters of a uniform tetradecahedron, having 2 congruent regular hexagonal faces, 12 congruent trapezoidal faces & 18 vertices lying on a spherical surface with certain radius, have been derived by the author H.C. Rajpoot by applying "HCR's Theory of Polygon" to calculate solid angle subtended by each regular hexagonal & trapezoidal face & their normal distances from the center of uniform tetradecahedron, inscribed radius, circumscribed radius, mean radius, surface area & volume. These formula are very useful in analysis, designing & modeling of various uniform polyhedra.


Mathematical Analysis Of Uniform Decahedron Having 10 Congruent Faces Each As A Right Kite By H.C. Rajpoot, Harish Chandra Rajpoot Rajpoot Hcr Feb 2015

Mathematical Analysis Of Uniform Decahedron Having 10 Congruent Faces Each As A Right Kite By H.C. Rajpoot, Harish Chandra Rajpoot Rajpoot Hcr

Harish Chandra Rajpoot H.C. Rajpoot

All the important parameters of a decahedron having 10 congruent faces each as a right kite have been derived by the author by applying HCR's Theory of Polygon to calculate normal distance of each face from the center, inscribed radius, circumscribed radius, mean radius, surface area & volume. The formula are very useful in analysis, designing & modeling of polyhedrons.


Mathematical Analysis Of Elliptical Path In The Annular Region Between Two Circles, Smaller Inside The Bigger One (Ellipse Between Two Circles By H.C. Rajpoot), Harish Chandra Rajpoot Rajpoot Hcr Feb 2015

Mathematical Analysis Of Elliptical Path In The Annular Region Between Two Circles, Smaller Inside The Bigger One (Ellipse Between Two Circles By H.C. Rajpoot), Harish Chandra Rajpoot Rajpoot Hcr

Harish Chandra Rajpoot H.C. Rajpoot

All the articles have been derived by the author by using simple geometry & trigonometry. These articles are related to the analysis of the elliptical path in the annular region between two circle, smaller inside bigger one & their centers separated by a certain distance. These formula are used to calculate minor axis, major axis, eccentricity & the radius of the third tangent circle touching the smaller circle externally & the bigger one internally. These articles (formula) are very practical & simple to apply in case studies & practical applications of 2-D Geometry.


Mathematical Derivations Of Inscribed & Circumscribed Radii For Three Externally Touching Circles (Geometry Of Circles By Hcr), Harish Chandra Rajpoot Hcr Feb 2015

Mathematical Derivations Of Inscribed & Circumscribed Radii For Three Externally Touching Circles (Geometry Of Circles By Hcr), Harish Chandra Rajpoot Hcr

Harish Chandra Rajpoot H.C. Rajpoot

All the articles, related to three externally touching circles, have been derived by using simple geometry & trigonometry to calculate inscribed & circumscribed radii. All the articles (formula) are very practical & simple to apply in case studies & practical applications of three externally touching circles in 2-D Geometry. Although these results are also valid in case of three spheres touching one another externally in 3-D geometry. These formula are also used for calculating any of three radii if rest two are known & the dimensions of the rectangle enclosing thee externally touching circles. Here is also the derivation of a general formula for computing ...


Mathematical Analysis Of Spherical Rectangle By H.C. Rajpoot, Harish Chandra Rajpoot Rajpoot Hcr Feb 2015

Mathematical Analysis Of Spherical Rectangle By H.C. Rajpoot, Harish Chandra Rajpoot Rajpoot Hcr

Harish Chandra Rajpoot H.C. Rajpoot

All the articles have been derived by Mr H.C. Rajpoot by using simple geometry & trigonometry. All the formula are very practical & simple to apply in case of any spherical rectangle to calculate all its important parameters such as solid angle, surface area covered, interior angles etc. & also useful for calculating all the parameters of the corresponding plane rectangle obtained by joining all the vertices of a spherical rectangle by the straight lines. These formula can also be used to calculate all the parameters of the right pyramid obtained by joining all the vertices of a spherical rectangle to the ...


Mathematical Analysis Of Spherical Triangle (Spherical Trigonometry By H.C. Rajpoot), Harish Chandra Rajpoot Rajpoot Hcr Feb 2015

Mathematical Analysis Of Spherical Triangle (Spherical Trigonometry By H.C. Rajpoot), Harish Chandra Rajpoot Rajpoot Hcr

Harish Chandra Rajpoot H.C. Rajpoot

All the important parameters of a spherical triangle have been derived by Mr H.C. Rajpoot by using simple geometry & trigonometry. All the articles (formula) are very practical & simple to apply in case of a spherical triangle to calculate all its important parameters such as solid angle, covered surface area, interior angles etc. & also useful for calculating all the parameters of the corresponding plane triangle obtained by joining all the vertices of a spherical triangle by straight lines. These formula can also be used to calculate all the parameters of the right pyramid obtained by joining all the vertices of ...