1) The document analyzes the mathematical properties of a truncated hexahedron (cube) which is generated by truncating the vertices of a regular cube.
2) It derives formulas to calculate key properties like edge length, normal distances, solid angles, volumes, surface area, radii and more using principles of right pyramids and HCR's theory of polygons.
3) Two methods are described for constructing a physical truncated hexahedron either from elementary right pyramids or by machining a solid sphere.
Mathematical analysis of a small rhombicuboctahedron (Archimedean solid) by HCRHarish Chandra Rajpoot
All the important parameters of a small rhombicuboctahedron (an Archimedean solid having 8 congruent equilateral triangular & 18 congruent square faces each of equal edge length) such as normal distances & solid angles subtended by the faces, inner radius, outer radius, mean radius, surface area & volume have been calculated by using HCR's formula for regular polyhedrons. This formula is a generalized dimensional formula which is applied on any of the five platonic solids i.e. reguler tetrahedron, regular hexahedron (cube), regular octahedron, regular dodecahedron & regular icosahedron to calculate their important parameters. It can also be used in analysis, designing & modelling of truncated polyhedrons.
HCR's Formula for Regular n-Polyhedrons (Mathematical analysis of regular n-p...Harish Chandra Rajpoot
This formula was derived by H.C. Rajpoot to calculate all the important parameters of a regular n-polyhedron such as inner radius, outer radius, mean radius, surface area & volume. This formula is a generalized dimensional formula which can be applied on any existing n-polyhedron since it depends on two parameters of any regular polyhedron as the no. of faces & the no. of edges in one face only. It can be used in analysis, designing & modelling of regular n-polyhedrons. It's named as "H. Rajpoot's Formula" by the author.
All the important parameters of a truncated dodecahedron (having 20 congruent equilateral triangular & 12 regular decagonal faces each of equal edge length) such as normal distances & solid angles subtended by the faces, inner radius, outer radius, mean radius, surface area & volume have been calculated by using HCR's formula for regular polyhedrons. This formula is a generalized dimensional formula which is applied on any of the five platonic solids i.e. reguler tetrahedron, regular hexahedron (cube), regular octahedron, regular dodecahedron & regular icosahedron to calculate their important parameters. It can be used in analysis, designing & modelling of polyhedrons.
Este documento contém 74 problemas de geometria espacial resolvidos, incluindo cálculos com cones, pirâmides, cilindros e esferas. As questões envolvem determinar medidas como volume, área total e lateral de sólidos geométricos usando suas propriedades e relações entre os elementos.
Este documento contém 212 problemas resolvidos de geometria espacial envolvendo cálculos de volumes, áreas e razões entre figuras geométricas como esferas, segmentos esféricos, calotas, fusos e zonas. As soluções exploram propriedades geométricas destas figuras para encontrar medidas desconhecidas a partir de informações fornecidas sobre raios, alturas, ângulos e distâncias.
A pirâmide quadrangular regular descrita tem uma base horizontal quadrada ABCD com 6cm de altura. Os vértices da base são A(1,1,2), B(-2,2,2), C, D e os vértices da pirâmide são indicados.
1. O documento apresenta uma lista de 25 exercícios sobre geometria plana de triângulos, incluindo cálculo de lados, ângulos e propriedades.
2. Fornece também as respostas para os exercícios.
3. O blog é dedicado a matemática para concursos e é administrado pelo professor Thieres Machado.
Mathematical analysis of a small rhombicuboctahedron (Archimedean solid) by HCRHarish Chandra Rajpoot
All the important parameters of a small rhombicuboctahedron (an Archimedean solid having 8 congruent equilateral triangular & 18 congruent square faces each of equal edge length) such as normal distances & solid angles subtended by the faces, inner radius, outer radius, mean radius, surface area & volume have been calculated by using HCR's formula for regular polyhedrons. This formula is a generalized dimensional formula which is applied on any of the five platonic solids i.e. reguler tetrahedron, regular hexahedron (cube), regular octahedron, regular dodecahedron & regular icosahedron to calculate their important parameters. It can also be used in analysis, designing & modelling of truncated polyhedrons.
HCR's Formula for Regular n-Polyhedrons (Mathematical analysis of regular n-p...Harish Chandra Rajpoot
This formula was derived by H.C. Rajpoot to calculate all the important parameters of a regular n-polyhedron such as inner radius, outer radius, mean radius, surface area & volume. This formula is a generalized dimensional formula which can be applied on any existing n-polyhedron since it depends on two parameters of any regular polyhedron as the no. of faces & the no. of edges in one face only. It can be used in analysis, designing & modelling of regular n-polyhedrons. It's named as "H. Rajpoot's Formula" by the author.
All the important parameters of a truncated dodecahedron (having 20 congruent equilateral triangular & 12 regular decagonal faces each of equal edge length) such as normal distances & solid angles subtended by the faces, inner radius, outer radius, mean radius, surface area & volume have been calculated by using HCR's formula for regular polyhedrons. This formula is a generalized dimensional formula which is applied on any of the five platonic solids i.e. reguler tetrahedron, regular hexahedron (cube), regular octahedron, regular dodecahedron & regular icosahedron to calculate their important parameters. It can be used in analysis, designing & modelling of polyhedrons.
Este documento contém 74 problemas de geometria espacial resolvidos, incluindo cálculos com cones, pirâmides, cilindros e esferas. As questões envolvem determinar medidas como volume, área total e lateral de sólidos geométricos usando suas propriedades e relações entre os elementos.
Este documento contém 212 problemas resolvidos de geometria espacial envolvendo cálculos de volumes, áreas e razões entre figuras geométricas como esferas, segmentos esféricos, calotas, fusos e zonas. As soluções exploram propriedades geométricas destas figuras para encontrar medidas desconhecidas a partir de informações fornecidas sobre raios, alturas, ângulos e distâncias.
A pirâmide quadrangular regular descrita tem uma base horizontal quadrada ABCD com 6cm de altura. Os vértices da base são A(1,1,2), B(-2,2,2), C, D e os vértices da pirâmide são indicados.
1. O documento apresenta uma lista de 25 exercícios sobre geometria plana de triângulos, incluindo cálculo de lados, ângulos e propriedades.
2. Fornece também as respostas para os exercícios.
3. O blog é dedicado a matemática para concursos e é administrado pelo professor Thieres Machado.
This document discusses engineering drawings and orthographic projections. It defines engineering drawings as drawings made by engineers using engineering tools, concepts and principles for engineering applications. It then explains orthographic projections as a way to represent 3D objects in 2D using multiple views. It outlines the principal planes of projection, principal views, positioning of objects in different quadrants, and conventions for first angle and third angle projections.
Bumi sebagai sfera - jarak pada permukaan bumiRosmaya Mokhtar
Tajuk Matematik Tingkatan 5 ; pengetahuan sedia ada - beza latitud, beza longitud, kedudukan tempat, bulatan agung ; boleh digunakan sebagai bahan bantu mengajar atau rujukan pelajar
Assalamualaikum
teman-teman ini adalah media pembelajaran interaktif materi kekongruenan bangun datar dan 2 segitiga untuk SMP kelas IX
Mudah-mudahan bermanfaat yaa :)
O documento contém uma revisão de matemática com exercícios sobre circunferências, raios, diâmetros e posições relativas entre circunferências. Também inclui exercícios sobre triângulos, perímetros e ângulos.
Kubus adalah bangun ruang sisi datar yang memiliki 6 sisi persegi, 8 titik sudut, 12 rusuk dan sisi diagonal yang sama panjang, serta 4 diagonal ruang.
Este documento contém 25 exercícios resolvidos sobre geometria espacial de sólidos como paralelepípedos, cubos, ortoedros e pirâmides. Os exercícios envolvem cálculos de áreas, volumes, dimensões e relações entre sólidos espaciais. As soluções apresentam os passos para chegar aos resultados requeridos em cada questão.
Dokumen tersebut membahas tentang kubus dan balok. Kubus dijelaskan memiliki 6 sisi yang sama besar dan semua sudutnya 90 derajat, sedangkan balok memiliki 3 pasang sisi yang sama besar namun minimal satu pasang berbeda ukuran. Unsur-unsur kubus dan balok juga dijelaskan seperti sisi, rusuk, titik sudut, diagonal bidang dan ruang, serta rumus volume dan luas permukaannya.
The document provides information about various plane curves covered in the Engineering Graphics course GE8152 Unit 1. It includes definitions and methods for constructing conic sections like circles, ellipses, parabolas and hyperbolas on a plane. It also describes construction of other curves like cycloids, epicycloids, hypocycloids, involutes of circles and squares. Detailed step-by-step processes are provided with diagrams to draw these curves using basic geometrical concepts like loci, tangents, normals and eccentricity.
Este documento apresenta resoluções de exercícios relacionados a cubos e paralelepípedos retângulos. São calculadas medidas como diagonais, áreas totais e volumes destes sólidos geométricos a partir de expressões algébricas envolvendo as dimensões dadas nos enunciados.
O documento discute trigonometria em triângulos retângulos. Explica como a trigonometria relaciona os lados e ângulos de um triângulo retângulo, introduz as razões trigonométricas e apresenta exemplos de cálculo delas. Também aborda identidades trigonométricas e aplicações em outros tipos de ângulos e arcos de circunferência.
O documento discute a história da geometria, classificação de polígonos e sólidos geométricos. Apresenta exemplos de sólidos como prisma triangular, pirâmide triangular e cubo, mostrando suas características como número de vértices, arestas e faces.
O documento apresenta 179 problemas de geometria espacial resolvidos, tratando de tópicos como volumes de sólidos gerados pela rotação de figuras planas, como triângulos, retângulos e polígonos, em torno de eixos. As soluções exploram propriedades geométricas como áreas, volumes, razões trigonométricas e proporcionalidade.
powerpoint ini membahas beberapa materi dari bidang kartesius, antara lain bentuk bidang kartesius, mendeskripsikan titik pada bidang katresius, menggambar titik pada bidang katresius, pengantar bidang koordinat, jarak, serta dilengkapi dengan latihan soal dan evaluasi.
Dokumen tersebut membahas tentang unsur-unsur lingkaran seperti titik pusat, jari-jari, diameter, busur, tali busur, tembereng, juring, dan apotema beserta rumus-rumus untuk menghitung keliling dan luas lingkaran.
Mathematical analysis of truncated octahedron (Applications of HCR's Theory)Harish Chandra Rajpoot
All the important parameters of a truncated octahedron such as normal distances & solid angles subtended by the faces, inner radius, outer radius, mean radius, surface area & volume have been calculated by using HCR's formula for regular polyhedron. This formula is a generalized dimensional formula which is applied on any of the five platonic solids i.e. reguler tetrahedron, regular hexahedron (cube), regular octahedron, regular dodecahedron & regular icosahedron to calculate their important parameters. It can be used in analysis, designing & modelling of regular n-polyhedrons.
All the important parameters of a cuboctahedron (Archimedean solid having 8 congruent equilateral triangular & 6 congruent square faces each of equal edge length) such as normal distances & solid angles subtended by the faces, inner radius, outer radius, mean radius, surface area & volume have been calculated by using HCR's formula for regular polyhedrons. This formula is a generalized dimensional formula which is applied on any of the five platonic solids i.e. reguler tetrahedron, regular hexahedron (cube), regular octahedron, regular dodecahedron & regular icosahedron to calculate their important parameters. It can also be used in analysis, designing & modelling of truncated polyhedrons.
This document discusses engineering drawings and orthographic projections. It defines engineering drawings as drawings made by engineers using engineering tools, concepts and principles for engineering applications. It then explains orthographic projections as a way to represent 3D objects in 2D using multiple views. It outlines the principal planes of projection, principal views, positioning of objects in different quadrants, and conventions for first angle and third angle projections.
Bumi sebagai sfera - jarak pada permukaan bumiRosmaya Mokhtar
Tajuk Matematik Tingkatan 5 ; pengetahuan sedia ada - beza latitud, beza longitud, kedudukan tempat, bulatan agung ; boleh digunakan sebagai bahan bantu mengajar atau rujukan pelajar
Assalamualaikum
teman-teman ini adalah media pembelajaran interaktif materi kekongruenan bangun datar dan 2 segitiga untuk SMP kelas IX
Mudah-mudahan bermanfaat yaa :)
O documento contém uma revisão de matemática com exercícios sobre circunferências, raios, diâmetros e posições relativas entre circunferências. Também inclui exercícios sobre triângulos, perímetros e ângulos.
Kubus adalah bangun ruang sisi datar yang memiliki 6 sisi persegi, 8 titik sudut, 12 rusuk dan sisi diagonal yang sama panjang, serta 4 diagonal ruang.
Este documento contém 25 exercícios resolvidos sobre geometria espacial de sólidos como paralelepípedos, cubos, ortoedros e pirâmides. Os exercícios envolvem cálculos de áreas, volumes, dimensões e relações entre sólidos espaciais. As soluções apresentam os passos para chegar aos resultados requeridos em cada questão.
Dokumen tersebut membahas tentang kubus dan balok. Kubus dijelaskan memiliki 6 sisi yang sama besar dan semua sudutnya 90 derajat, sedangkan balok memiliki 3 pasang sisi yang sama besar namun minimal satu pasang berbeda ukuran. Unsur-unsur kubus dan balok juga dijelaskan seperti sisi, rusuk, titik sudut, diagonal bidang dan ruang, serta rumus volume dan luas permukaannya.
The document provides information about various plane curves covered in the Engineering Graphics course GE8152 Unit 1. It includes definitions and methods for constructing conic sections like circles, ellipses, parabolas and hyperbolas on a plane. It also describes construction of other curves like cycloids, epicycloids, hypocycloids, involutes of circles and squares. Detailed step-by-step processes are provided with diagrams to draw these curves using basic geometrical concepts like loci, tangents, normals and eccentricity.
Este documento apresenta resoluções de exercícios relacionados a cubos e paralelepípedos retângulos. São calculadas medidas como diagonais, áreas totais e volumes destes sólidos geométricos a partir de expressões algébricas envolvendo as dimensões dadas nos enunciados.
O documento discute trigonometria em triângulos retângulos. Explica como a trigonometria relaciona os lados e ângulos de um triângulo retângulo, introduz as razões trigonométricas e apresenta exemplos de cálculo delas. Também aborda identidades trigonométricas e aplicações em outros tipos de ângulos e arcos de circunferência.
O documento discute a história da geometria, classificação de polígonos e sólidos geométricos. Apresenta exemplos de sólidos como prisma triangular, pirâmide triangular e cubo, mostrando suas características como número de vértices, arestas e faces.
O documento apresenta 179 problemas de geometria espacial resolvidos, tratando de tópicos como volumes de sólidos gerados pela rotação de figuras planas, como triângulos, retângulos e polígonos, em torno de eixos. As soluções exploram propriedades geométricas como áreas, volumes, razões trigonométricas e proporcionalidade.
powerpoint ini membahas beberapa materi dari bidang kartesius, antara lain bentuk bidang kartesius, mendeskripsikan titik pada bidang katresius, menggambar titik pada bidang katresius, pengantar bidang koordinat, jarak, serta dilengkapi dengan latihan soal dan evaluasi.
Dokumen tersebut membahas tentang unsur-unsur lingkaran seperti titik pusat, jari-jari, diameter, busur, tali busur, tembereng, juring, dan apotema beserta rumus-rumus untuk menghitung keliling dan luas lingkaran.
Mathematical analysis of truncated octahedron (Applications of HCR's Theory)Harish Chandra Rajpoot
All the important parameters of a truncated octahedron such as normal distances & solid angles subtended by the faces, inner radius, outer radius, mean radius, surface area & volume have been calculated by using HCR's formula for regular polyhedron. This formula is a generalized dimensional formula which is applied on any of the five platonic solids i.e. reguler tetrahedron, regular hexahedron (cube), regular octahedron, regular dodecahedron & regular icosahedron to calculate their important parameters. It can be used in analysis, designing & modelling of regular n-polyhedrons.
All the important parameters of a cuboctahedron (Archimedean solid having 8 congruent equilateral triangular & 6 congruent square faces each of equal edge length) such as normal distances & solid angles subtended by the faces, inner radius, outer radius, mean radius, surface area & volume have been calculated by using HCR's formula for regular polyhedrons. This formula is a generalized dimensional formula which is applied on any of the five platonic solids i.e. reguler tetrahedron, regular hexahedron (cube), regular octahedron, regular dodecahedron & regular icosahedron to calculate their important parameters. It can also be used in analysis, designing & modelling of truncated polyhedrons.
Mathematical analysis of truncated tetrahedron (Application of HCR's Theory o...Harish Chandra Rajpoot
All the important parameters of a truncated tetrahedron such as normal distances & solid angles subtended by the faces, inner radius, outer radius, mean radius, surface area & volume have been calculated by using HCR's formula for regular polyhedron. This formula is a generalized dimensional formula which is applied on any of the five platonic solids i.e. reguler tetrahedron, regular hexahedron (cube), regular octahedron, regular dodecahedron & regular icosahedron to calculate their important parameters. It can be used in analysis, designing & modelling of regular n-polyhedrons.
Mathematical analysis of truncated icosahedron & identical football by applyi...Harish Chandra Rajpoot
All the important parameters of a truncated icosahedron (Goldberg polyhedron, G(1,1)) such as normal distances & solid angles of faces, inner radius, outer radius, mean radius, surface area & volume have been calculated by using HCR's formula for regular polyhedron. This formula is a generalized dimensional formula which is applied on any of the five platonic solids i.e. reguler tetrahedron, regular hexahedron (cube), regular octahedron, regular dodecahedron & regular icosahedron to calculate their important parameters. It depends on two parameters of any regular polyhedron as the no. of faces & the no. of edges in one face only. It is also used for finding out all the important parameters of a football similar to a truncated icosahedron. It can be used in analysis, designing & modelling of regular n-polyhedrons.
Mathematical analysis of small rhombicosidodecahedron (Archimedean solid) by hcrHarish Chandra Rajpoot
This document provides a mathematical analysis of the small rhombicosidodecahedron, an Archimedean solid. It begins by describing how the solid can be generated from a regular icosahedron or dodecahedron. It then uses HCR's formulas for regular polyhedrons to derive equations for key parameters of the small rhombicosidodecahedron, including the angle between faces, outer radius, normal distances of different faces from the center, and solid angles subtended by each face. The analysis shows that the pentagonal faces are closest to the center and define the inner radius of the solid.
Mathematical analysis of great rhombicuboctahedron (an Archimedean solid) (Ap...Harish Chandra 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 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 are very useful in analysis, designing & modeling of various uniform polyhedra.
Mathematical analysis of decahedron with 10 congruent faces each as a right k...Harish Chandra 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 n-gonal trapezohedron with 2n congruent right kite f...Harish Chandra Rajpoot
The generalized formula are applicable on any n-gonal trapezohedronhaving 2n congruent right kite faces, 4n edges & 2n+2 vertices lying on a spherical surface with a certain radius. These formula have been derived by the author Mr H.C. Rajpoot to analyse infinite no. of the trapezohedrons having congruent right kite faces simply by setting n=3,4,5,6,7,………………upto infinity, to calculate all the important parameters such as ratio of unequal edges, outer radius, inner radius, mean radius, surface area, volume, solid angles subtended by the polyhedron at its vertices, dihedral angles between the adjacent right kite faces etc. These formula are very useful for the analysis, modeling & designing of various n-gonal trapezoherons/deltohedrons.
Mathematical Analysis of Rhombicuboctahedron (Application of HCR's Theory)Harish Chandra Rajpoot
The author H C Rajpoot has derived the radius of circumscribed sphere passing through all 24 identical vertices of a rhombicuboctahedron with given edge length by applying ‘HCR’s Theory of Polygon’ & subsequently derived various formula to analytically compute the normal distances of equilateral triangular & square faces from the centre of rhombicuboctahedron, radius of mid-sphere, surface area, volume, solid angles subtended by each equilateral triangular face & each square face at the centre by using ‘HCR’s Theory of Polygon’, dihedral angle between each two faces meeting at any of 24 identical vertices (i.e. truncated rhombic dodecahedron), solid angle subtended by truncated rhombic dodecahedron at any of its 24 identical vertices.
Mathematical analysis of great rhombicosidodecahedron (the largest Archimedea...Harish Chandra 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 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. These formula are very useful in analysis, designing & modeling of various uniform polyhedrons.
Mathematical Analysis of Regular Spherical Polygons (Application of HCR's The...Harish Chandra Rajpoot
This document presents a mathematical analysis of regular spherical polygons. It derives a characteristic equation relating the radius of the sphere, length of each side, interior angle, and number of sides. This equation can be used to calculate any parameter when the other three are known. The document also discusses calculating the solid angle, area, and other properties of both regular spherical polygons and their corresponding planar polygons. It provides examples of applying the analysis to polygons on both general spheres and the unit sphere.
Mathematical analysis of truncated rhombic dodecahedron (HCR's Polyhedron)Harish Chandra Rajpoot
1) The document describes H.C. Rajpoot's analysis of a truncated rhombic dodecahedron using his "Theory of Polygon".
2) Key results include formulas for the radius of the circumscribed sphere, normal distances to different face types, surface area, volume, and dihedral angles between faces.
3) The analysis involves deriving the truncated polyhedron from a rhombic dodecahedron and establishing relationships between their geometric properties.
The author has derived the formula to analytically compute all the important parameters of a disphenoid (isosceles tetrahedron with four congruent acute-triangular faces) such as volume, surface area, vertical height, radii of inscribed & circumscribed spheres, solid angle subtended at each vertex, coordinates of vertices, in-centre, circum-centre & centroid of a disphenoid for the optimal configuration in 3D space. The author has also proved the important conclusions related to a disphenoid by mathematical derivations using 3D coordinate geometry.
Mathematical Analysis & Modeling of Pyramidal Flat Container, Right Pyramid &...Harish Chandra Rajpoot
This document derives generalized formulas to compute important parameters like the V-cut angle, edge length, dihedral angle, surface area, and volume of pyramidal flat containers, pyramids, and polyhedrons with regular polygonal bases. It applies HCR's Theorem and Corollary to develop formulas for a pyramidal container with a regular n-gonal base in terms of the base side length, slant height, and face inclination angle. Steps for constructing paper models of such containers are also outlined.
Mathematical Analysis of Tetrahedron (dihedral angles between the consecutive...Harish Chandra 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 regular & uniform polyhedrons to calculate the solid angle subtended by polyhedron at its vertex.
Mathematical Analysis of Rhombic Dodecahedron by applying HCR's Theory of Pol...Harish Chandra Rajpoot
In this paper, the author Mr H C Rajpoot mathematically analyses & derives analytic formula for a rhombic dodecahedron having 12 congruent faces each as a rhombus, 24 edges & 14 vertices lying on a spherical surface with a certain radius. ‘HCR’s Theory of Polygon’ is used to derive formula to analytically compute the angles & diagonals of rhombic face, radii of circumscribed sphere, inscribed sphere & midsphere, surface area & volume of rhombic dodecahedron in terms of edge length, solid angles subtended at the vertices and dihedral angle between adjacent faces. This convex polyhedron can be constructed by joining 12 congruent elementary-right pyramids with rhombic base & certain normal height.
Mathematical analysis of sphere resting in the vertex of polyhedron, filletin...Harish Chandra 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 distance of sphere from the vertex, distance of sphere from the edges, fillet radius of the faces etc. The formula have been generalized for packing the spheres in the vertices of right pyramids & all five platonic solids.
Mathematical Analysis of Spherical Rectangle (Application of HCR's Theory of ...Harish Chandra 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 center of the sphere such as normal height, angle between the consecutive lateral edges, area of plane rectangular base etc.
All the standard formula from 'Advanced Geometry' by the author Mr H.C. Rajpoot have been included in this book. These formula are related to the solid geometry dealing with the 2-D & the figures in 3-D space & miscellaneous articles in Trigonometry & Geometry. These are useful the standard formula for case studies & practical applications.
All the standard formula from 'Advanced Geometry' by the author Mr H.C. Rajpoot have been included in this book. These formula are related to the solid geometry dealing with the 2-D & 3-D figures in the space & miscellaneous articles in Trigonometry & Geometry. These are useful the standard formula to be remembered for case studies & practical applications. Although, all the formula for the plane figures (i.e. planes bounded by the straight lines only) can be derived by using standard formula of right triangle that has been explained in details in "HCR's Theory of Polygon" published with International Journal of Mathematics & Physical Sciences Research in Oct, 2014.
And the analysis of oblique frustum of right circular cone has been explained in his research paper 'HCR's Infinite-series' published with IJMPSR
Similar to Mathematical analysis of truncated cube/hexahedron (20)
Mathematical analysis of a non-uniform tetradecahedron having 2 congruent reg...Harish Chandra Rajpoot
All the important parameters of a non-uniform tetradecahedron, having 2 congruent regular hexagonal faces, 12 congruent trapezoidal faces & 18 vertices lying on a spherical surface with a certain radius, have been derived by the author 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 non-uniform tetradecahedron, inscribed radius, circumscribed radius, mean radius, surface area & volume. These formulas are very useful in the analysis, designing & modeling of various non-uniform polyhedra.
Mathematical analysis of non-uniform polyhedra having 2 congruent regular n-g...Harish Chandra Rajpoot
All the important formulas have been generalized which are applicable to calculate the important parameters, of any non-uniform polyhedron having 2 congruent regular n-gonal faces, 2n congruent trapezoidal faces each with three equal sides, 5n edges & 3n vertices lying on a spherical surface, such as solid angle subtended by each face at the centre, normal distance of each face from the centre, inner radius, outer radius, mean radius, surface area & volume. These are useful for the analysis, designing & modeling of different non-uniform polyhedra.
Regular N-gonal Right Antiprism: Application of HCR’s Theory of PolygonHarish Chandra Rajpoot
A regular n-gonal right antiprism is a semiregular convex polyhedron which has 2n identical vertices all lying on a sphere, 4n edges, and (2n+2) faces out of which 2 are congruent regular n-sided polygons, and 2n are congruent equilateral triangles such that all the faces have equal side. The equilateral triangular faces meet the regular polygonal faces at the common edges and vertices alternatively such that three equilateral triangular faces meet at each of 2n vertices. This paper presents, in details, the mathematical derivations of the generalized and analytic formula which are used to determine the different important parameters in terms of edge length, such as normal distances of faces, normal height, radius of circumscribed sphere, surface area, volume, dihedral angles between adjacent faces, solid angle subtended by each face at the centre, and solid angle subtended by polygonal antiprism at each of its 2n vertices using HCR’s Theory of Polygon. All the generalized formulae have been derived using simple trigonometry, and 2D geometry which are difficult to derive using any other methods.
The document presents mathematical derivations of parameters for a regular pentagonal right antiprism. It derives analytic formulas for the antiprism's normal heights, normal distances from faces to the center, radius of the circumscribed sphere, surface area, volume, and other values in terms of the antiprism's edge length. All formulas are derived using trigonometry and 2D geometry applied to the transformation of a regular icosahedron into the antiprism shape.
Derivation of great-circle distance formula using of HCR's Inverse cosine for...Harish Chandra Rajpoot
The author derives the great-circle distance formula using hcr's inverse cosine formula. An analytic and the most generalized formula has been derived to accurately compute the minimum distance or great circle distance between any two arbitrary points on a sphere of finite radius which is equally applicable in the geometry of sphere. This formula is extremely useful to calculate the geographical distance between any two points on the globe for the given latitudes & longitudes. This formula is the most power tool which is applicable for all the distances on the tiny sphere as well as the large sphere like giant planet assuming them the perfect spheres.
The circumscribed and the inscribed polygons are well known and mathematically well defined in the context of 2D-Geometry. The term ‘Circum-inscribed Polygon’ has been proposed by the author and used as a new definition of the polygon which satisfies the conditions of a circumscribed polygon and an inscribed polygon together. In other words, the circum-inscribed polygon is a polygon which has both the inscribed and circumscribed circles. The newly defined circum-inscribed polygon has each of its sides touching a circle and each of its vertices lying on another circle. The most common examples of circum-inscribed polygon are triangle, regular polygon, trapezium with each of its non-parallel sides equal to the Arithmetic Mean (AM) of its parallel sides (called circum-inscribed trapezium) and right kite. This paper describes the mathematical derivations of the analytic formula to find out the different parameters in terms of AM and GM of known sides such as radii of circumscribed & inscribed circles, unknown sides, interior angles, diagonals, angle between diagonals, ratio of intersecting diagonals, perimeter, area, and distance between circum-centre and in-centre of circum-inscribed trapezium. Like an inscribed polygon, a circum-inscribed polygon always has all of its vertices lying on infinite number of spherical surfaces. All the analytic formulae have been derived using simple trigonometry and 2-dimensional geometry which can be used to analyse the complex 2D and 3D geometric figures such as cyclic quadrilateral and trapezohedron, and other polyhedrons.
HCR's theorem (Rotation of two co-planar planes, meeting at angle bisector, a...Harish Chandra Rajpoot
In this theorem, the author derives a mathematical expression to analytically compute the V-cut angle (δ) required for rotating through the same angle (θ) the two co-planar planes, initially meeting at a common edge bisecting the angle (α) between their intersecting straight edges, about their intersecting straight edges until their new straight edges (generated after removing V-shaped planar region) coincide. As a result, we get a point (apex) where three planes intersect one another out of which two are original planes (rotated) & third one is their co-plane (fixed).
This theorem is very important for creating pyramidal flat containers with polygonal (regular or irregular) base, closed right pyramids & polyhedrons having two regular n-gonal & 2n congruent trapezoidal faces.
The author has also presented some paper models of pyramidal flat containers with regular pentagonal, heptagonal and octagonal bases
How to compute area of spherical triangle given the aperture angles subtended...Harish Chandra Rajpoot
The author Mr H.C. Rajpoot has derived the general formula to compute the area of the spherical triangle having each side as a great circle arc on the spherical surface when 1.) aperture angle subtended by each of three sides at the center of sphere are known 2.) arc length of each of three sides is known. These formula are applicable for any spherical triangle to the compute area on the sphere.
This document summarizes 9 formulas derived by Harish Chandra Rajpoot related to geometry of 2D shapes like squares, triangles, trapezoids, and circles. The derivations use basic geometry and trigonometry. Formula 1 finds the angle subtended by a point inside a square. Formula 2 finds the area of a quadrilateral formed inside a square. Subsequent formulas find radii or lengths related to combinations of these basic shapes. Diagrams and step-by-step workings are provided for each formula derivation.
Mathematical derivations of some important formula in 2D-Geometry H.C. RajpootHarish Chandra Rajpoot
Here are some important formula in 2D-Geometry derived by the author Mr H.C. Rajpoot using simple geometry & trigonometry. The formula, derived here, are related to the triangle, square, trapezium & tangent circles. These formula are very useful for case studies in 2D-Geometry to compute the important parameters of 2D-figures.
Volume & surface area of right circular cone cut by a plane parallel to its s...Harish Chandra Rajpoot
All the articles have been derived by the author by using simple geometry, trigonometry & calculus. All the formula are the most generalized expressions which can be used for computing the volume & surface area of minor & major parts usually each with hyperbolic section obtained by cutting a right circular cone with a plane parallel to its symmetrical (longitudinal) axis.
This formula holds good for all the regular spherical polygons. It is a very important formula (mathematical relation) applicable on any regular spherical polygon having each of its sides as an arc of the great circle on a spherical surface. It is of crucial importance to find out any of the four important parameters (i.e. radius of sphere, no. of sides, length of side, interior angle of polygon) if other three are given (known) for any regular spherical polygon. It also concludes that any three of the four parameters are self-sufficient to exactly represent a unique regular spherical polygon. A regular spherical polygon having three known parameters can be created or drawn only on a unique spherical surface of a radius which is easily found out by HCR's formula.
Mathematical analysis of identical circles touching one another on the spheri...Harish Chandra 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 spherical surfaces.
Mathematical analysis of identical circles touching one another on the whole ...Harish Chandra 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 designing & modeling all five platonic solids having identical flat circular faces.
Reflection of a point about a line & a plane in 2-D & 3-D co-ordinate systems...Harish Chandra Rajpoot
The document discusses methods for calculating the reflection of points about lines and planes in 2D and 3D coordinate systems. It provides general expressions to find the coordinates of a reflected point given a point and line/plane of reflection. For a point reflected about a line or plane, the midpoint between the original and reflected point must lie on the line/plane and the line connecting the points must be perpendicular/parallel to the line/plane. Using geometry relationships, the author derives formulas to calculate the coordinates of the reflected point and foot of the perpendicular from the original point to the line/plane.
Derivations of inscribed & circumscribed radii for three externally touching ...Harish Chandra 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 the length of common chord of two intersecting circles.
Mathematical Analysis of Spherical Triangle (Spherical Trigonometry by H.C. R...Harish Chandra 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 a spherical triangle to the center of sphere such as normal height, angle between the consecutive lateral edges, area of base etc.
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UPRAHUL
This Dissertation explores the particular circumstances of Mirzapur, a region located in the
core of India. Mirzapur, with its varied terrains and abundant biodiversity, offers an optimal
environment for investigating the changes in vegetation cover dynamics. Our study utilizes
advanced technologies such as GIS (Geographic Information Systems) and Remote sensing to
analyze the transformations that have taken place over the course of a decade.
The complex relationship between human activities and the environment has been the focus
of extensive research and worry. As the global community grapples with swift urbanization,
population expansion, and economic progress, the effects on natural ecosystems are becoming
more evident. A crucial element of this impact is the alteration of vegetation cover, which plays a
significant role in maintaining the ecological equilibrium of our planet.Land serves as the foundation for all human activities and provides the necessary materials for
these activities. As the most crucial natural resource, its utilization by humans results in different
'Land uses,' which are determined by both human activities and the physical characteristics of the
land.
The utilization of land is impacted by human needs and environmental factors. In countries
like India, rapid population growth and the emphasis on extensive resource exploitation can lead
to significant land degradation, adversely affecting the region's land cover.
Therefore, human intervention has significantly influenced land use patterns over many
centuries, evolving its structure over time and space. In the present era, these changes have
accelerated due to factors such as agriculture and urbanization. Information regarding land use and
cover is essential for various planning and management tasks related to the Earth's surface,
providing crucial environmental data for scientific, resource management, policy purposes, and
diverse human activities.
Accurate understanding of land use and cover is imperative for the development planning
of any area. Consequently, a wide range of professionals, including earth system scientists, land
and water managers, and urban planners, are interested in obtaining data on land use and cover
changes, conversion trends, and other related patterns. The spatial dimensions of land use and
cover support policymakers and scientists in making well-informed decisions, as alterations in
these patterns indicate shifts in economic and social conditions. Monitoring such changes with the
help of Advanced technologies like Remote Sensing and Geographic Information Systems is
crucial for coordinated efforts across different administrative levels. Advanced technologies like
Remote Sensing and Geographic Information Systems
9
Changes in vegetation cover refer to variations in the distribution, composition, and overall
structure of plant communities across different temporal and spatial scales. These changes can
occur natural.
How to Make a Field Mandatory in Odoo 17Celine George
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Executive Directors Chat Leveraging AI for Diversity, Equity, and InclusionTechSoup
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How to Setup Warehouse & Location in Odoo 17 InventoryCeline George
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This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
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This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.