[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-297437-105":59,"doc-detail-297437-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","121-introduction-surface-areas-and-volumes-of-combinations-of-solids","12.1 Introduction - Surface Areas and Volumes of Combinations of Solids","","The text introduces how to find surface areas and volumes of composite solids formed by combining two or more basic solids such as cylinders, cones, and hemispheres. It motivates the topic using everyday examples like a tank-like container with hemispherical ends and a test tube shape. The chapter then explains computing total surface area by summing relevant curved and exposed areas, while excluding parts that are joined together, and illustrates the method through multiple worked examples with formulas and numerical results.",{"@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":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/121-introduction-surface-areas-and-volumes-of-combinations-of-solids/297437/",{"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/121-introduction-surface-areas-and-volumes-of-combinations-of-solids/297437.png","ImageObject",300,407,{"name":92,"@type":93},"Rhys","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-30","2026-09-18",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",15,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"Why can’t these composite objects be classified under the basic solids studied earlier?","Question",{"text":113,"@type":114},"Composite objects combine two or more basic solids, and their exposed surfaces don’t match the earlier single-solid categories. Therefore, the surface area and volume must be handled by breaking the shape into parts and considering only the exposed areas.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"How is the total surface area of a combination of solids computed for a cylinder with two hemispheres?",{"text":118,"@type":114},"The total surface area is obtained by adding the curved surface areas of the two hemispheres and the curved surface area of the cylinder. Joined flat faces are not included; only the curved parts that remain exposed are counted.",{"name":120,"@type":111,"acceptedAnswer":121},"In Example 2 (cube with a hemisphere on top), why is the cube’s area under the hemisphere excluded?",{"text":122,"@type":114},"The part of the cube where the hemisphere is attached is covered and not exposed to paint or counting for surface area. Hence, that base contact area is subtracted before adding the hemisphere’s curved surface area.","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},297437,1789744535,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":19,"category_name":20,"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":52,"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":130,"read_time":144},687207024643,"https://ap-avatar.wpscdn.com/davatar_3d24733baf745e90a7e4bdd5f77d97b2","1062CH13  \n# 12.1 Introduction\n\nFrom Class IX,you are familiar with some of the solids like cuboid,cone,cylinder,andsphere(see Fig.12.1).You have also learnt how to find their surface areas and volumes.  \n(i)  \nFig.12.1  \nInour day-to-day life,we come across a number of solids made upof 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.12.2),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(ii)(iii)(iv)  \nFig.12.2  \nAgain,you may have seen an object like theone in Fig.12.3.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 a combination 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# 12.2 Surface Area of a Combination of Solids\n\nLet us consider the container seen in Fig.12.2.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.12.4,after we put the pieces all together.  \nFig.12.4  \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.12.3  \n   \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.12.5.  \nFig.12.5  \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 colourit with his crayons.The topisshaped like a cone surmounted by a hemisphere(see Fig 12.6).The entire top is 5 cm in height andthe diameter of the top is 3.5cm.Find the area he  \nhas to colour   \nSolution:This top is exactly like the object we have discussed in Fig.12.5.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 surface area of the hemisphere  \n   \nFig.12.6  \nAlso,the height of the cone=height of the top-height(radius)of the hemispherical part  \ns tesmnentecosmopmoc)  \nTherefore,  \nThis gives the surface area of the top as  \nYou may note that 'total surface area of the top'is not the sum of the total  \nsurface areas of the cone and hemisphere.  \nExample 2:The decorative block shownin Fig.12.7 is made of two solids ","cbCaioFuv7uJoGo6","https://ap.wps.com/l/cbCaioFuv7uJoGo6","pdf",1620005,"English","# 12.1 Introduction\n# 12.2 Surface Area of a Combination of Solids\n## Example 1: Cone surmounted by a hemisphere\n## Example 2: Cube with a hemisphere on top\n## Example 3: Cone mounted on a cylinder\n## Example 4: Cylinder with a hemispherical depression","[{\"question\":\"Why can’t these composite objects be classified under the basic solids studied earlier?\",\"answer\":\"Composite objects combine two or more basic solids, and their exposed surfaces don’t match the earlier single-solid categories. Therefore, the surface area and volume must be handled by breaking the shape into parts and considering only the exposed areas.\"},{\"question\":\"How is the total surface area of a combination of solids computed for a cylinder with two hemispheres?\",\"answer\":\"The total surface area is obtained by adding the curved surface areas of the two hemispheres and the curved surface area of the cylinder. Joined flat faces are not included; only the curved parts that remain exposed are counted.\"},{\"question\":\"In Example 2 (cube with a hemisphere on top), why is the cube’s area under the hemisphere excluded?\",\"answer\":\"The part of the cube where the hemisphere is attached is covered and not exposed to paint or counting for surface area. Hence, that base contact area is subtracted before adding the hemisphere’s curved surface area.\"}]","12.1 Introduction - Surface Areas and Volumes of Combinations of Solids | PDF",25]