[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-297574-105":59,"doc-detail-297574-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","6b-volume-and-surface-area-of-3d-shapes-a-level-links","6b - Volume and surface area of 3D shapes - A Level links","","A Level learning material on volume and surface area of 3D shapes, covering key formulas for prisms, pyramids, cylinders, spheres and cones. Includes worked examples for calculating missing dimensions, such as prism length and total volume of a composite solid using π. Provides a practice set with problems on spheres, cuboids, prisms and hemispheres, plus extension tasks involving deriving expressions from diagrams and comparing volumes via surface-area conditions.",{"@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/6b-volume-and-surface-area-of-3d-shapes-a-level-links/297574/",{"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/6b-volume-and-surface-area-of-3d-shapes-a-level-links/297574.png","ImageObject",300,407,{"name":92,"@type":93},"Angel","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-03","2026-09-18",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",26,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"What is the formula for the volume of a prism?","Question",{"text":113,"@type":114},"The volume of a prism equals cross-sectional area multiplied by length.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"How do you calculate the total surface area of a cylinder?",{"text":118,"@type":114},"Total surface area of a cylinder is 2πr² + 2πrh.",{"name":120,"@type":111,"acceptedAnswer":121},"How can you find a missing length in a triangular prism problem?",{"text":122,"@type":114},"Use the given volume to substitute into the triangular prism volume formula, then simplify and rearrange to solve for the length, keeping units consistent.","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},297574,1789745864,{"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":24,"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},687207412472,"https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d","Volume and surface area of 3D shapes  \nA LEVEL LINKS  \nScheme of work: 6b. Gradients, tangents, normals, maxima and minima  \nKey points  \n􀁸 Volume of a prism = cross-sectional area × length.  \n􀁸 The surface area of a 3D shape is the total area of all its faces.  \n􀁸 Volume of a pyramid = 13 × area of base × vertical height.  \n􀁸 Volume of a cylinder = πr2h  \n􀁸 Total surface area of a cylinder = 2πr2 + 2πrh  \n􀁸 Volume of a sphere = 43 􀁓 r3  \n􀁸 Surface area of a sphere = 4πr2  \n􀁸 Volume of a cone = 13 􀁓 r2 h  \n􀁸 Total surface area of a cone = πrl + πr2  \nExamples  \nExample 1  \nThe triangular prism has volume 504 cm3. Work out its length.  \n| V = ~~1~~2 bhl\u003Cbr>504 = ~~1~~2 × 9 × 4 × l\u003Cbr>504 = 18 × ll = 504 ÷ 18 = 28 cm | 1 Write out the formula for the volume of a triangular prism.\u003Cbr>2 Substitute known values into the formula.\u003Cbr>3 Simplify\u003Cbr>4 Rearrange to work out l.\u003Cbr>5 Remember the units. |\n| --- | --- |\n\nExample 2 Calculate the volume of the 3D solid.  \nGive your answer in terms of π.  \n| Total volume = volume of hemisphere + Volume of cone\u003Cbr>= ~~1~~2 of ~~4~~3 πr3 + ~~1~~3 πr2h\u003Cbr> 1   4\u003Cbr>Total volume = 2 × 3 × π × 53\u003Cbr> 1\u003Cbr>+ 3 × π × 52 × 7\u003Cbr>425\u003Cbr>= 3 π cm3 | 1 The solid is made up ofa hemisphere radius 5 cm anda cone with radius 5 cm and height\u003Cbr>12 − 5 = 7 cm.\u003Cbr>2 Substitute the measurements into the formula for the total volume.\u003Cbr>3 Remember the units. |\n| --- | --- |\n\nPractice  \n1 Work out the volume of each solid.  \nLeave your answers in terms of π where appropriate.  \na  \nb  \nc  \ne  \nd  \nf  \na sphere with radius 7 cm  \ng a sphere with diameter 9 cm h a hemisphere with radius 3 cm  \ni j  \n2 A cuboid has width 9.5 cm, height 8 cm and volume 1292 cm3. Work out its length.  \n3 The triangular prism has volume 1768 cm3. Work out its height.  \nExtend  \n4 The diagram shows a solid triangular prism.  \nAll the measurements are in centimetres. The volume of the prism is V cm3.  \nFind a formula for Vin terms of x.  \nGive your answer in simplified form.  \n5 The diagram shows the area of each of three faces ofa cuboid.  \nThe length of each edge of the cuboid is a whole number of centimetres.  \nWork out the volume of the cuboid.  \n6 The diagram shows a large catering size tin of beans in the shape of a cylinder.  \nThe tin has a radius of 8 cm and a height of 15 cm. A company wants to make a new size of tin.  \nThe new tin will have a radius of 6.7 cm.  \nIt will have the same volume as the large tin.  \nCalculate the height of the new tin.  \nGive your answer correct to one decimal place.  \n7 The diagram shows a sphere and a solid cylinder.  \nThe sphere has radius 8 cm.  \nThe solid cylinder has a base radius of 4 cm anda height of h cm.  \nThe total surface area of the cylinder is half the total surface area of the sphere.  \nWork out the ratio of the volume of the sphere to the volume of the cylinder.  \nGive your answer in its simplest form.  \n8 The diagram shows a solid metal cylinder.  \nThe cylinder has base radius 4x and height 3x.  \nThe cylinder is melted down and made into a sphere of radius r.  \nFind an expression for r in terms of x.  \nAnswers  \n1 a V = 396 cm3 b V = 75 000 cm3 c V = 402.5 cm3 d V = 200π cm3 e V = 1008π cm3 f V= 13732 π cm3 g V = 121.5π cm3 h V = 18π cm3 i V = 48π cm3 j V = 983 π cm3  \n2 17 cm  \n3 17 cm  \n4 V = x3 + 172 x2 + 4x  \n5 60 cm3  \n6 21.4 cm  \n7 32 : 9  \n8 r 􀀠 36x","cbCaimDLbE8xEvJu","https://ap.wps.com/l/cbCaimDLbE8xEvJu","pdf",469194,"English","# Scheme of work\n## Key points\n## Examples\n## Practice\n## Extend\n## Answers","[{\"question\":\"What is the formula for the volume of a prism?\",\"answer\":\"The volume of a prism equals cross-sectional area multiplied by length.\"},{\"question\":\"How do you calculate the total surface area of a cylinder?\",\"answer\":\"Total surface area of a cylinder is 2πr² + 2πrh.\"},{\"question\":\"How can you find a missing length in a triangular prism problem?\",\"answer\":\"Use the given volume to substitute into the triangular prism volume formula, then simplify and rearrange to solve for the length, keeping units consistent.\"}]","6b - Volume and surface area of 3D shapes - A Level links | PDF",13]