[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-297472-105":59,"doc-detail-297472-en":130},{"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":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","131-introduction-surface-areas-and-volumes-of-combinations-of-solids","13.1 Introduction - Surface Areas and Volumes of Combinations of Solids","","Chapter 13 explains how to find surface areas and volumes of solids formed by combining two or more basic solids already studied. It motivates the need by real-life examples such as a container with a cylinder and hemispheres and an object like a test tube made from a cylinder and a hemisphere. Section 13.2 develops the approach: break the composite solid into simpler parts, sum only the exposed curved/required surfaces, and use surface area formulas for combinations illustrated through worked examples.",{"@graph":69,"@context":122},[70,84,105],{"@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":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/131-introduction-surface-areas-and-volumes-of-combinations-of-solids/297472/",{"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/131-introduction-surface-areas-and-volumes-of-combinations-of-solids/297472.png","ImageObject",300,407,{"name":92,"@type":93},"Gelato","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-20","2026-09-18",true,{"@type":102,"interactionType":103,"userInteractionCount":44},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"Why do we need a new method for surface area and volume in this chapter?","Question",{"text":112,"@type":113},"The solids formed in daily life are combinations of basic solids, and their surface areas/volumes cannot be directly classified as any single solid studied earlier.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How is the total surface area of a combination solid found?",{"text":117,"@type":113},"Break the solid into simpler known shapes, then add only the surface areas that are actually exposed (e.g., curved surfaces of hemispheres and the cylinder).",{"name":119,"@type":110,"acceptedAnswer":120},"In a hemisphere–cone toy, why are the flat faces not included?",{"text":121,"@type":113},"The toy is formed by joining the cone and hemisphere at their flat faces, so that joined area is not exposed and is excluded from the painted surface area.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},297472,1789744828,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":44,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":129,"read_time":144},19241457091524,"https://us-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c","# 13.1 Introduction\n\nFrom Class IX,you are familiar with some of the solids like cuboid,cone,cylinder,andsphere(see Fig.13.1).You have also learnt how to find their surface areas and volumes.  \n(i)  \n(ii)  \nFig.13.1  \nIn our day-to-day life,we come across a number of solids madeupof combinationsof two or more of the basic solids as shown above.  \nYou must have seen a truck with acontainer fitted on its back(see Fig.13.2),carrying oil or water from one place toanother.Is it in the shape of any of the fourbasic solids mentioned above?You mayguess that it is made of a cylinder with twohemispheres as its ends  \n(iii)  \n(iv)  \nFig.13.2  \nAgain,you may have seen an object like theone in Fig.13.3.Can you name it?Atest tube,right!You would have used one in your science laboratory.This tube is also a combination of a cylinder and ahemisphere.Similarly,while travelling,you may haveseen some big and beautiful buildings or monumentsmade up of acombination of solids mentioned above.  \nIf for some reason you wanted to find thesurface areas,or volumes,or capacities of suchobjects,how would you do it?We cannot classifythese under any of the solids you have already studied.  \nIn this chapter,you will see how to find surface areas and volumes of suchobjects.  \n# 13.2 Surface Area of a Combination of Solids\n\nLet us consider the container seen in Fig.13.2.How do we find the surface area ofsuch a solid?Now,whenever we come across a new problem,we first try to see,ifwe can break it down into smaller problems,we have earlier solved.We can see thatthis solid is made up of a cylinder with two hemispheres stuck at either end.It wouldlook like what we have in Fig.13.4,after we put the pieces all together.  \nFig.13.4  \nIf we consider the surface of the newly formed object,we would be able to seeonly the curved surfaces of the two hemispheres and the curved surface of the cylinder.  \nSo,the total surface area of the new solid is the sum of the curved surfaceareas of each of the individual parts.This gives,  \nTSA of new solid=CSA of one hemisphere +CSA of cylinder  \n+CSA of other hemisphere  \nwhere TSA,CSA stand for ‘Total Surface Area'and ‘Curved Surface Area'respectively.  \nLet us now consider another situation.Suppose we are making a toy by puttingtogether a hemisphere and a cone.Let us see the steps that we would be goingthrough.  \nFig.13.3  \nFirst,we would take a cone and a hemisphere and bring their flat faces together.Here,of course,we would take the base radius of the cone equal to the radius of thehemisphere,for the toy is to have a smooth surface.So,the steps would be as shownin Fig.13.5.  \nFig.13.5  \nAt the end of our trial,we have got ourselves a nice round-bottomed toy.Now ifwe want to find how much paint we would require to colour the surface of this toy,what would we need to know?We would need to know the surface area of the toy,which consists of the CSA of the hemisphere and the CSA of the cone.  \nSo,we can say:  \nTotal surface area of the toy =CSA of hemisphere+CSA of coneNow,let us consider some examples.  \nExample1:Rasheed got a playing top(lattu)as hisbirthday present,which surprisingly had no colour onit.He wanted to colour it with his crayons.The topisshaped like a cone surmounted by a hemisphere(see Fig 13.6).The entire top is 5cm in height andthe diameter of the top is 3.5 cm.Find the area he  \nhas to colour (Take π;  \nSolution:This top is exactly like the object we have discussed in Fig.13.5.So,we  \ncan conveniently use the result we have arrived at there.That is:  \nTSA of the toy=CSA of hemisphere+CSA of cone  \nNow,the curved suface area of the hemisphere  \nFig.13.6  \nAlso,the height of the cone=height of the top-height(radius)of the hemispherical part  \ns.tesutegtftecoe  \nTherefore,  \nThis gives the surface area of the top as  \nsurface areas of the cone and hemisphere.  \nExample 2:The decorative block shownin Fig.13.7 is made of two solids —a cubeand a hemisphere.The base of the blockis acube with edge 5 cm,and the hemispherefixe","cbCaio9ujTv5fPqI","https://ap.wps.com/l/cbCaio9ujTv5fPqI","pdf",752256,21,"English","# 13.1 Introduction\n# 13.2 Surface Area of a Combination of Solids\n## Cylinder with two hemispheres\n## Toy formed by hemisphere and cone\n## Worked examples","[{\"question\":\"Why do we need a new method for surface area and volume in this chapter?\",\"answer\":\"The solids formed in daily life are combinations of basic solids, and their surface areas/volumes cannot be directly classified as any single solid studied earlier.\"},{\"question\":\"How is the total surface area of a combination solid found?\",\"answer\":\"Break the solid into simpler known shapes, then add only the surface areas that are actually exposed (e.g., curved surfaces of hemispheres and the cylinder).\"},{\"question\":\"In a hemisphere–cone toy, why are the flat faces not included?\",\"answer\":\"The toy is formed by joining the cone and hemisphere at their flat faces, so that joined area is not exposed and is excluded from the painted surface area.\"}]","13.1 Introduction - Surface Areas and Volumes of Combinations of Solids | PDF",53]