[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-297512-105":59,"doc-detail-297512-en":131},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":124,"head_meta":126,"extra_data":128,"updated_unix":130},105,"en","hcrs-infinite-series-analysis-of-oblique-frustum-of-a-right-circular-cone-derivation-of-volume-and-surface-area","HCR’s Infinite-series - Analysis of Oblique Frustum of a Right Circular Cone - Derivation of Volume and Surface Area","","Three infinite series were derived to compute the volume, surface area, and related geometric quantities of the elliptical-section oblique frustum formed by cutting a right circular cone with an inclined plane. The method uses double integration in polar coordinates, binomial expansion, and -functions, addressing the lack of direct mathematical formulas for the frustum’s volume and surface measures. Each series is a discrete summation of infinite terms that converges to finite values, named HCR’s convergence series.",{"@graph":69,"@context":123},[70,84,106],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/hcrs-infinite-series-analysis-of-oblique-frustum-of-a-right-circular-cone-derivation-of-volume-and-surface-area/297512/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/hcrs-infinite-series-analysis-of-oblique-frustum-of-a-right-circular-cone-derivation-of-volume-and-surface-area/297512.png","ImageObject",300,407,{"name":92,"@type":93},"Emma Wilson","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-04","2026-09-18",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",14,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"What geometric object is analyzed in the document?","Question",{"text":113,"@type":114},"An oblique frustum with an elliptical section is generated by cutting a right circular cone with a plane inclined relative to the cone’s axis and located at a distance from the apex.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"How are the volume and surface area expressions derived?",{"text":118,"@type":114},"The derivation uses double integration in polar coordinates, binomial expansion, and -functions, then expresses results as discrete summations of infinite terms that converge.",{"name":120,"@type":111,"acceptedAnswer":121},"Why are the results called HCR’s convergence series?",{"text":122,"@type":114},"The derived series are written as infinite discrete summations whose terms converge to finite values, enabling practical computations for the frustum parameters.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},297512,1789745232,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":140,"language":141,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":142,"faqs":143,"seo_title":144,"seo_description":67,"update_tm":130,"read_time":145,"preview_page_count":4},3848291630094,"https://eur-avatar.wpscdn.com/davatar_085a072bc5b1113ac321206ff7593b45","“HCR’s Infinite-series”Analysis of Oblique Frustum of a Right  \nCircular Cone  \nMr Harish Chandra Rajpoot  \nMadan Mohan Malaviya University of Technology, Gorakhpur-273010 (UP) India  \nAbstract: Three series had been derived by the author, using double-integration in polar co-ordinates, binomial  \nexpansion and -functions for determining the volume, surface-area & perimeter of elliptical-section of  \noblique frustum of a right circular cone as there had not been any mathematical formula for determining the same due to some limitations. All these three series are in form of discrete summation of infinite terms which converge into finite values hence these were also named as HCR’s convergence series. These are extremely useful in case studies & practical computations.  \nKeywords: HCR’s Convergence-series, oblique frustum, binomial expansion, -functions.  \nI. INTRODUCTION  \nWhen a right circular cone is thoroughly cut by a plane inclined at a certain angle with the axis of a right circular cone, an oblique frustum with elliptical section & apex point is generated So far there are no mathematical formulae for determining the volume & surface area of oblique frustum & perimeter, major axis, minor axis & eccentricity of elliptical section. This article derives mathematical formulae for all the parameters.  \nII. VOLUME OF OBLIQUE FRUSTUM  \nLet there be a right circular cone with apex angle Now, it is thoroughly cut by a plane (as shown by the extended line AB in fig 1.) inclined at an angle with axis OZ of the cone & lying at a normal distance h from the apex point O  \nFig 1: Oblique frustum of right cone with elliptical section  \nNow, consider any parametric point ( ) on the conical surface Equation of conical surface: in Cartesian co-ordinates  \n( )  \nIn Polar co-ordinates  \n( )  \n( )  \nEquation of cutting plane: by intercept form  \n~~ ~~ ~~ ~~ ~~ ~~ ⇒  \nEquation of elliptical boundary: Ellipse is the curve of intersection of conical surface & cutting plane. Hence on substituting the co-ordinates of parametric point ( ) in the equation of plane, we have the following equation in polar co-ordinates  \n* ( ) +  \n*  \n+  \n⇒  \n( )  \n( )  \nNow, let‟s consider an imaginary cylindrical surface ABDC completely enclosing the frustum of solid cone (as shown in the fig 2. below) . It is easier to calculate the volume of void space enclosed by the frustum of cone, cylindrical surface & XY-plane.  \nFig 2: Imaginary cylindrical surface enclosing the frustum of cone  \nLet‟s consider an elementary surface area on the conical surface (see fig 1.)  \n* ( ) +  \nTaking the projection of elementary area on XY-plane  \n⇒ ( )  \nNow, volume of elementary vertical strip  \n⇒ ( )  \nHence, using double integration with proper limits, total volume ( ) of the enclosed void space  \n⇒ ∫ ∫∫∫  \n∫  \n∫  \n( )  \n~~ ~~ ∫  \n∫  \n*∫ + ( )  \n*~~ ~~+ ~~ ~~ ∫( ), we have (( )  )  \n( ) . .~~ ~~ / /  \n∫  \n. .~~ ~~ / /( ( ) )  \n( )  \nSince, the frustum has finite elliptical section, hence we have a condition  \n~~ ~~ ⇒ ~~ ~~  \n| ( ) |  \nNow, using Binomial Expansion  \n( ) ( )  ( ) ( ) ( | | ) We have,  \n⇒ ~~ ~~ ∫ ( ) (  )  \n~~ ~~ ∫ [ ( ) ~~ ~~ ( ) ~~ ~~ ( ) ~~ ~~ ( )  \n~~ ~~ ( ) ]  \n~~ ~~ *∫ ~~ ~~ ∫ ~~ ~~ ∫  \n~~ ~~ ∫ ~~ ~~ ∫  \n~~ ~~ ∫ +  \n(Multiplying & diving each term by 2 in above series)  \n~~ ~~ * ~~ ~~ ∫ ~~ ~~ ∫ ~~ ~~ ∫  \n~~ ~~ ∫ ~~ ~~ ∫ +  \n∫ ( ) ( ) ( )  \n∫ ( ) ( ) ( )  \n∫ ∫ ∫  \n∫ ∫ ∫  \nNow, we have  \n~~ ~~ [ ~~ ~~ ∫ ~~ ~~ ∫ ~~ ~~ ∫  \n]  \n~~ ~~ [ ~~ ~~ ∫ ~~ ~~ ∫ ~~ ~~ ∫ ]  \n~~ ~~ [ ~~ ~~ ∫ ~~ ~~ ∫ ~~ ~~ ∫ ]  \nSince, we know from -functions  \n∫ ~~ ~~.~~ ~~.~~  ~~/~~ ~~.~~  ~~/~~ ~~/ ( ) √~~ ~~  \nHence, we have  \n~~ ~~ 0 ~~ ~~ .~~ ~~.~~ ~~.~~ ~~/~~ ~~.~~  ~~/~~ ~~// ~~ ~~ .~~ ~~.~~ ~~.~~ ~~/~~ ~~.~~  ~~/~~ ~~// ~~ ~~ .~~ ~~.~~ ~~.~~ ~~/~~ ~~.~~  ~~/~~ ~~// 1  \n~~ ~~ 0 ~~ ~~ .~~ ~~.~~ ~~/(~~ ~~).~~ ~~// ~~ ~~ .~~ ~~.~~ ~~/(~~ ~~).~~ ~~// ~~ ~~ .~~ ~~.~~ ~~/(~~ ~~).~~ ~~// 1  \n0 ~~ . ~~√~~ ~~√~~ ~~/ ~~ .  ~~√~~ ~~√~~ ~~/ ~~ .  ~~√~~ ~~√~~ ~~/ 1  \n0 ~~ ~~ ~~ ~~ .~~   ~~ / ~~ ~~ ~~ ","cbCaigJ2lYTN2vMk","https://ap.wps.com/l/cbCaigJ2lYTN2vMk","pdf",1117997,17,"English","# I. INTRODUCTION\n# II. VOLUME OF OBLIQUE FRUSTUM\n# III. SURFACE AREA OF OBLIQUE FRUSTUM","[{\"question\":\"What geometric object is analyzed in the document?\",\"answer\":\"An oblique frustum with an elliptical section is generated by cutting a right circular cone with a plane inclined relative to the cone’s axis and located at a distance from the apex.\"},{\"question\":\"How are the volume and surface area expressions derived?\",\"answer\":\"The derivation uses double integration in polar coordinates, binomial expansion, and -functions, then expresses results as discrete summations of infinite terms that converge.\"},{\"question\":\"Why are the results called HCR’s convergence series?\",\"answer\":\"The derived series are written as infinite discrete summations whose terms converge to finite values, enabling practical computations for the frustum parameters.\"}]","HCR’s Infinite-series - Analysis of Oblique Frustum of a Right Circular Cone - Derivation of Volume and Surface Area | PDF",43]