[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-297256-105":3,"detail-sidebar-cat-0-en-105":81,"doc-detail-297256-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","cuboid-investigation-surface-area-and-volume-study","Cuboid Investigation - Surface Area and Volume Study","","Cuboid Investigation explores relationships between cuboid dimensions, surface area, and volume. It starts from standard formulas for volume and total surface area across all six faces, then focuses on cuboids with volume 900. The task is to find the largest and smallest possible surface areas, using factor triples for the side lengths and scaling methods to generate new dimension sets. Solutions conclude that more similar edge lengths reduce surface area, approaching the minimum near a cube, and list many integer factor triples with their surface areas.",{"@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/research-report/","Research & Report",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/document/cuboid-investigation-surface-area-and-volume-study/297256/",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/cuboid-investigation-surface-area-and-volume-study/297256.png","ImageObject",300,407,{"name":42,"@type":43},"Aurora","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 formulas define a cuboid’s volume and surface area?","Question",{"text":63,"@type":64},"Volume equals length × width × height. Total surface area equals 2 × (lw + lh + wh), covering all six faces.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"How do you begin the investigation for cuboids with volume 900?",{"text":68,"@type":64},"Choose any three numbers whose product is 900 and treat them as the cuboid’s dimensions. Use the surface area formula to compute the total surface area.",{"name":70,"@type":61,"acceptedAnswer":71},"How does changing dimensions affect the surface area?",{"text":72,"@type":64},"After you find one valid dimension triple, you can form new triples by scaling one dimension up while scaling another down. The results show that making dimensions more similar tends to lower the surface area, with the minimum occurring near a cube.","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},297256,1789743399,{"code":4,"msg":82,"data":83},"success",[84,88,92,96,101,106,111,115,120,123,127],{"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":93,"show_sort_weight":94,"slug":95},"Exam",70,"exam",{"id":97,"doc_module":4,"doc_module_name":25,"category_name":98,"show_sort_weight":99,"slug":100},5,"Comic",60,"comic",{"id":102,"doc_module":4,"doc_module_name":25,"category_name":103,"show_sort_weight":104,"slug":105},6,"Technology",50,"technology",{"id":107,"doc_module":4,"doc_module_name":25,"category_name":108,"show_sort_weight":109,"slug":110},7,"Healthcare",40,"healthcare",{"id":112,"doc_module":4,"doc_module_name":25,"category_name":29,"show_sort_weight":113,"slug":114},8,30,"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":97,"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":112,"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":80,"read_time":112,"preview_page_count":4},4810365810221,"https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d","Cuboid Investigation  \n\n| Formula: | 􀜸 = 􀝈􀝁􀝊􀝃􀝐ℎ × 􀝓􀝅􀝀􀝐ℎ × ℎ􀝁􀝅􀝃ℎ􀝐 |\n| --- | --- |\n| Units:  Usually 􀜿􀝉3 (“cubic centimetres”,“centimetres cubed”,“cc”,“ml”) Surface area:  The total area (flat space) of all 6 faces of the cuboid |  |\n| Formula: | 􀜵􀜣 = 2 × (􀝈􀝁􀝊􀝃􀝐ℎ × 􀝓􀝅􀝀􀝐ℎ + 􀝈􀝁􀝊􀝃􀝐ℎ × ℎ􀝁􀝅􀝃ℎ􀝐 + 􀝓􀝅􀝀􀝐ℎ × ℎ􀝁􀝅􀝃ℎ􀝐) |\n| Units: | Usually 􀜿􀝉2 (“square centimetres”,“centimetres squared”) |\n\nVolume:  Total space inside the cuboid  \nInvestigate the different surface areas of cuboids with a volume of 􀫢􀫙􀫙􀢉􀢓􀫜 . What is the largest surface area you can find? What is the smallest?  \n􀁸 How do I begin?  \nThink of any three numbers that multiply together to make 900. These can be the dimensions of your cuboid. Draw a diagram and work out the total surface area.  \n􀁸 What next?  \nOnce you have three numbers that work, you can create a new set of three by scaling one number up and another down. Eg, if you have 50􀜿􀝉, 9􀜿􀝉 and 2􀜿􀝉, then 25􀜿􀝉, 18􀜿􀝉 and 2􀜿􀝉 will also work (halving one number and doubling another in this case) .  \nEg:  \nDimensions: 􀜮􀝁􀝊􀝃􀝐ℎ = 2􀜿􀝉 􀜹􀝅􀝀􀝐ℎ = 50􀜿􀝉 􀜪􀝁􀝅􀝃ℎ􀝐 = 9􀜿􀝉  \nSurface area: 2 × (2 × 50 + 2 × 9 + 50 × 9) = 􀫚􀫚􀫜􀫟􀢉􀢓􀫛  \nConclusions:  \nThe cuboid with the largest surface area was:    \nIt has a total surface area of:    \nThe cuboid with the smallest surface area was:    \nIt has a total surface area of:    \nWhat do you notice?  \nCuboid Investigation SOLUTIONS  \n\n| Formula: | 􀜸 = 􀝈􀝁􀝊􀝃􀝐ℎ × 􀝓􀝅􀝀􀝐ℎ × ℎ􀝁􀝅􀝃ℎ􀝐 |\n| --- | --- |\n| Units:  Usually 􀜿􀝉3 (“cubic centimetres”,“centimetres cubed”,“cc”,“ml”) Surface area:  The total area (flat space) of all 6 faces of the cuboid |  |\n| Formula: | 􀜵􀜣 = 2 × (􀝈􀝁􀝊􀝃􀝐ℎ × 􀝓􀝅􀝀􀝐ℎ + 􀝈􀝁􀝊􀝃􀝐ℎ × ℎ􀝁􀝅􀝃ℎ􀝐 + 􀝓􀝅􀝀􀝐ℎ × ℎ􀝁􀝅􀝃ℎ􀝐) |\n| Units: | Usually 􀜿􀝉2 (“square centimetres”,“centimetres squared”) |\n\nVolume:  Total space inside the cuboid  \nInvestigate the different surface areas of cuboids with a volume of 􀫢􀫙􀫙􀢉􀢓􀫜 . What is the largest surface area you can find? What is the smallest?  \n􀁸 How do I begin?  \nThink of any three numbers that multiply together to make 900. These can be the dimensions of your cuboid. Draw a diagram and work out the total surface area.  \n􀁸 What next?  \nOnce you have three numbers that work, you can create a new set of three by scaling one number up and another down. Eg, if you have 50􀜿􀝉, 9􀜿􀝉 and 2􀜿􀝉, then 25􀜿􀝉, 18􀜿􀝉 and 2􀜿􀝉 will also work (halving one number and doubling another in this case) .  \nEg:  \nDimensions: 􀜮􀝁􀝊􀝃􀝐ℎ = 2􀜿􀝉 􀜹􀝅􀝀􀝐ℎ = 50􀜿􀝉 􀜪􀝁􀝅􀝃ℎ􀝐 = 9􀜿􀝉  \nSurface area: 2 × (2 × 50 + 2 × 9 + 50 × 9) = 􀫚􀫚􀫜􀫟􀢉􀢓􀫛  \nConclusions:  \nThe cuboid with the largest surface area was: 1􀜿􀝉 × 1􀜿􀝉 × 900􀜿􀝉  \nIt has a total surface area of: 3602􀜿􀝉2  \n(note: no limit on surface area if not restricted to whole numbers)  \nThe cuboid with the smallest surface area was: 9􀜿􀝉 × 10􀜿􀝉 × 10􀜿􀝉  \nIt has a total surface area of: 560􀜿􀝉2  \n(note: the cube 9 .65 …􀜿􀝉 × 9 .65 …􀜿􀝉 × 9 .65 …􀜿􀝉 × gives the overall min: 559 .3 …􀜿􀝉2 )  \nWhat do you notice?  \nThe more similar the lengths, the smaller the surface area.  \nThe closer the cuboid gets to a cube, the smaller the surface area.  \nCuboid Investigation SOLUTIONS Continued  \nFull list of possible surface areas for each integer factor triple:  \n\n| Length | Width | Height | SA |\n| --- | --- | --- | --- |\n| 9 | 10 | 10 | 560 |\n| 6 | 10 | 15 | 600 |\n| 5 | 12 | 15 | 630 |\n| 5 | 10 | 18 | 640 |\n| 5 | 9 | 20 | 650 |\n| 6 | 6 | 25 | 672 |\n| 4 | 15 | 15 | 690 |\n| 5 | 6 | 30 | 720 |\n| 4 | 9 | 25 | 722 |\n| 5 | 5 | 36 | 770 |\n| 3 | 15 | 20 | 810 |\n| 3 | 12 | 25 | 822 |\n| 3 | 10 | 30 | 840 |\n| 4 | 5 | 45 | 850 |\n| 3 | 6 | 50 | 936 |\n| 3 | 5 | 60 | 990 |\n| 2 | 18 | 25 | 1072 |\n| 3 | 4 | 75 | 1074 |\n| 2 | 15 | 30 | 1080 |\n| 2 | 10 | 45 | 1120 |\n| 2 | 9 | 50 | 1136 |\n| 3 | 3 | 100 | 1218 |\n| 2 | 6 | 75 | 1224 |\n| 2 | 5 | 90 | 1280 |\n| 2 | 3 | 150 | 1512 |\n| 2 | 2 | 225 | 1808 |\n| 1 | 30 | 30 | 1920 |\n| 1 | 25 | 36 | 1922 |\n| 1 | 20 | 45 | 1930 |\n| 1 | 18 | 50 | 1936 |\n| 1 | 15 | 60 | 1950 |\n| 1 | 12 | 75 | 1974 |\n| 1 | 10 | 90 | 2000 |\n| 1 | 9 | 100 | 2018 |\n| 1 | 6 | 150 | 2112 |\n| 1 | 5 | 180 | 2170 |\n| 1 |","cbCaintui2W7HKKk","https://ap.wps.com/l/cbCaintui2W7HKKk","pdf",440393,"English","# Cuboid Investigation\n## Surface area and volume formulas\n## Investigate cuboids with volume 900\n## How to begin and scale dimensions\n## Conclusions and observations\n## Solutions: surface area results for factor triples","[{\"question\":\"What formulas define a cuboid’s volume and surface area?\",\"answer\":\"Volume equals length × width × height. Total surface area equals 2 × (lw + lh + wh), covering all six faces.\"},{\"question\":\"How do you begin the investigation for cuboids with volume 900?\",\"answer\":\"Choose any three numbers whose product is 900 and treat them as the cuboid’s dimensions. Use the surface area formula to compute the total surface area.\"},{\"question\":\"How does changing dimensions affect the surface area?\",\"answer\":\"After you find one valid dimension triple, you can form new triples by scaling one dimension up while scaling another down. The results show that making dimensions more similar tends to lower the surface area, with the minimum occurring near a cube.\"}]","Cuboid Investigation - Surface Area and Volume Study | PDF"]