[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-297246-105":3,"detail-sidebar-cat-0-en-105":81,"doc-detail-297246-en":131},{"code":4,"msg":5,"data":6},0,"ok",{"site_id":7,"language":8,"slug":9,"title":10,"keywords":11,"description":12,"schema_data":13,"social_meta":74,"head_meta":76,"extra_data":78,"updated_unix":80},105,"en","surface-areas-volumes-year-8-10-areas-and-volumes-of-regular-and-composite-solids","Surface-areas-volumes-year-8-10 - Areas and volumes of regular and composite solids","","A structured math lesson for Years 8 to 10 on surface area and volume of regular and composite 3D solids. It introduces key formulae for spheres, cylinders, and cones, including total surface area and component-based calculations. The lesson explains how minor and major domes relate to a sphere when the minor dome subtends 120° at the centre. Worked examples combine frustums, domes, and cylinders to compute curved areas, TSA, and volumes, followed by an exercise to find TSA and volume for composite solids.",{"@graph":14,"@context":73},[15,34,56],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/exam/","Exam",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/document/surface-areas-volumes-year-8-10-areas-and-volumes-of-regular-and-composite-solids/297246/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":49,"encodingFormat":47,"isAccessibleForFree":50,"interactionStatistic":51},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/surface-areas-volumes-year-8-10-areas-and-volumes-of-regular-and-composite-solids/297246.png","ImageObject",300,407,{"name":42,"@type":43},"Jiven","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-11-04","2026-09-18",true,{"@type":52,"interactionType":53,"userInteractionCount":55},"InteractionCounter",{"@type":54},"ViewAction",11,{"@type":57,"mainEntity":58},"FAQPage",[59,65,69],{"name":60,"@type":61,"acceptedAnswer":62},"What are the surface area formulae for a sphere, cylinder, and cone?","Question",{"text":63,"@type":64},"The lesson gives SA for a sphere as SA = 4πr². For a cylinder it uses TSA = 2πr² + 2πrl, and for a cone it uses TSA = πr² + πrs (base plus curved surface).","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"How are the minor dome and major dome defined in the sphere examples?",{"text":68,"@type":64},"They are defined for a sphere of radius r so that the minor dome subtends an angle of 120° at the centre. This allows the lesson to compute minor dome area as πr² and major dome area as 3πr².",{"name":70,"@type":61,"acceptedAnswer":71},"How do you calculate the TSA of composite solids in the worked examples?",{"text":72,"@type":64},"The lesson forms TSA by adding the relevant components, such as a minor dome plus a frustum of a cone and a base, or two major domes plus a cylinder’s curved surface. Each example substitutes the given measurements into the surface area formulae.","https://schema.org",{"og:url":32,"og:type":75,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":77,"canonical":32},"index,follow",{"doc_id":79,"site_id":7},297246,1790521365,{"code":4,"msg":82,"data":83},"success",[84,88,92,95,100,105,110,115,120,123,127],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":85,"show_sort_weight":86,"slug":87},"Story & Novel",90,"story-novel",{"id":26,"doc_module":4,"doc_module_name":25,"category_name":89,"show_sort_weight":90,"slug":91},"Literature",80,"literature",{"id":33,"doc_module":4,"doc_module_name":25,"category_name":29,"show_sort_weight":93,"slug":94},70,"exam",{"id":96,"doc_module":4,"doc_module_name":25,"category_name":97,"show_sort_weight":98,"slug":99},5,"Comic",60,"comic",{"id":101,"doc_module":4,"doc_module_name":25,"category_name":102,"show_sort_weight":103,"slug":104},6,"Technology",50,"technology",{"id":106,"doc_module":4,"doc_module_name":25,"category_name":107,"show_sort_weight":108,"slug":109},7,"Healthcare",40,"healthcare",{"id":111,"doc_module":4,"doc_module_name":25,"category_name":112,"show_sort_weight":113,"slug":114},8,"Research & Report",30,"research-report",{"id":116,"doc_module":4,"doc_module_name":25,"category_name":117,"show_sort_weight":118,"slug":119},9,"Religion & Spirituality",20,"religion-spirituality",{"id":118,"doc_module":4,"doc_module_name":25,"category_name":121,"show_sort_weight":118,"slug":122},"World Cup","world-cup",{"id":124,"doc_module":4,"doc_module_name":25,"category_name":125,"show_sort_weight":124,"slug":126},10,"Lifestyle","lifestyle",{"id":128,"doc_module":4,"doc_module_name":25,"category_name":129,"show_sort_weight":96,"slug":130},19,"General","general",{"code":4,"msg":82,"data":132},{"doc_id":79,"user_id":133,"nickname":42,"user_avatar":134,"doc_module":4,"category_id":33,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":55,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":30,"language":140,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":12,"update_tm":144,"read_time":111},1099513958607,"https://ap-avatar.wpscdn.com/avatar/100002390cf8733938c?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778829742770036399","Math Lesson (Suitable for Years 8 to 10) Areas and volumes of regular and composite solids  \n© itute 2018  \nSurface area formulae  \nSphere: SA = 4πr2  \n(SA: surface area)  \nCylinder:  \nTwo ends + curved surface TSA = 2πr2 + 2πrl  \n= 2πr(r + l)  \n(TSA: total surface area)  \nCone: Base + curved surface TSA = πr2 + πrs  \nr  \nl r  \ns  \nr  \nDo you know the following facts about a sphere of radius r?  \nThe above is true only when the minor dome subtendsan angle of 120o at the centre of the sphere.  \nThe following examples involve the use of these two particular domes, a cylinder and a cone to form composite solids.  \nPreliminary: Consider a sphere of radius 23 units. It is divided into a major dome and a minor dome such that the minor dome subtends an angle of 120o at the centre of the sphere.  \nThe triangle in the above drawing is similar to the special 30-60-90 triangle shown below. It is enlarged by a scale factor of 3 .  \nMultiply each side of the 30-60-90 triangle by 3  \n 3 × 3 = 3   \n30o   \n1 × 3 = 3  \n2 × 3 = 23  60o  \nExample 1 Find the area of the curved surface of the frustum of the cone. (Measurements in cm)  \nArea = π rs (large cone)– π rs (small cone)= π (6)(12)– π (3)(6) = 90π cm2  \nExample 2 Find the surface areas of the minor dome and the major dome. (Measurements in cm)  \nThe radius of the sphere is 23 cm (refer to preliminary) .  \nArea of minor dome = π r2 = π (23 )2 = 12π cm2 Area of major dome = 3π r2 = 3π(23 )2 = 36π cm2  \nAreas and volumes of regular and composite solids 1 © itute 2018  \nExample 3 Find the TSA of the composite solid formed by placing the minor dome in Example 2 ontop of the frustum of the cone in Example 1.  \nTSA  \n= minor dome + frustum of cone + base = 12π + 90π + π (6) 2  \n= 138π cm2  \nExample 4 Find the TSA of the composite solid formed by placing a major dome discussed in Example 2 on each end of a 10 cm long cylinder.  \nTSA  \n= 2 major domes + cylindrical curved surface = 2 × 36π + 2π rl  \n= 2 × 36π + 2π(3)(10)  \n= 132π cm2  \nVolume formulae  \nSphere: V = ~~ ~~43 π r3  \nCylinder: V = π r2l  \nCone: V = ~~ ~~13π r2h  \nl  \nh  \nr  \nr  \nDo you know the following facts about a sphere of radius r?  \nThe above is true only when the minor dome subtendsan angle of 120o at the centre of the sphere.  \nExample 5 Find the volume of the frustum of the cone. (Measurements are in cm)  \nThe heights of the cones were obtained by the method outlined in the preliminary, using scale factors 3 and 6.  \nVolume  \n= large cone – small cone  \n= 1 π (6) 2 (6~~ ~~3 ) −  1 π (3) 2 (3 3 ) = 63 3 π cm3  \n3 3  \nExample 6 Find the volumes of the minor dome and the major dome. (Measurements are in cm)  \nMinor dome:  \nV =  5  × 􀀆􀀄 4 π r3 􀀃􀀁 =  5  × π(23 )3 = 53 π 32 􀀅3 􀀂 24  \nMajor dome:  \nV =  27 × 􀀆􀀄 4 π r3 􀀃􀀁 = 9 × π(23 )3 = 273 π 32 􀀅3 􀀂 8  \ncm3  \ncm3  \nAreas and volumes of regular and composite solids 2 © itute 2018  \nExample 7 Find the volume of material to be cut from an appropriate cone to form the composite solid discussed in Example 3.  \nVolume of material to be removed  \n= volume of small cone – volume of minor dome = 1 π (3)2 (3 3 )− 53 π = 4~~ ~~3 π cm3  \n3  \nExample 8 Find the volume of material to be cut from an appropriate cylinder to form the composite solid discussed in example 4.  \n33   10  33   \nRadius of the large cylinder = 23 cm Length of the large cylinder = 63 +10 cm  \nVolume of material to be removed  \n= Large cylinder – small cylinder – 2 major domes = π (23 )2 (63 + 10)− π(3)2 (10) − 2(273 π )  \n= 6π(33 + 5) cm3  \nExercise  \nFind the TSA and volume of each of the following three-dimensional composite solids.  \n(1)  \n(2)  \n(3)  \nAreas and volumes of regular and composite solids 3 © itute 2018","cbCaihpEJK28KHFs","https://ap.wps.com/l/cbCaihpEJK28KHFs","pdf",132353,"English","# Surface area formulae\n## Sphere, cylinder, cone\n## Composite solids using domes, frustums, and cylinders\n# Volume formulae\n## Sphere, cylinder, cone\n## Composite-solid volumes\n# Exercise\n## Find TSA and volume for composite solids","[{\"question\":\"What are the surface area formulae for a sphere, cylinder, and cone?\",\"answer\":\"The lesson gives SA for a sphere as SA = 4πr². For a cylinder it uses TSA = 2πr² + 2πrl, and for a cone it uses TSA = πr² + πrs (base plus curved surface).\"},{\"question\":\"How are the minor dome and major dome defined in the sphere examples?\",\"answer\":\"They are defined for a sphere of radius r so that the minor dome subtends an angle of 120° at the centre. This allows the lesson to compute minor dome area as πr² and major dome area as 3πr².\"},{\"question\":\"How do you calculate the TSA of composite solids in the worked examples?\",\"answer\":\"The lesson forms TSA by adding the relevant components, such as a minor dome plus a frustum of a cone and a base, or two major domes plus a cylinder’s curved surface. Each example substitutes the given measurements into the surface area formulae.\"}]","Surface-areas-volumes-year-8-10 - Areas and volumes of regular and composite solids | PDF",1789743357]