33:I[196793,["/_next/static/chunks/0j4ivqk4ne18h.js?dpl=nogit-1791235224","/_next/static/chunks/2_6olli-sutu3.js?dpl=nogit-1791235224","/_next/static/chunks/0zcakt1-boe_f.js?dpl=nogit-1791235224","/_next/static/chunks/0psy-n5us57a4.js?dpl=nogit-1791235224","/_next/static/chunks/3mz725-y91ph6.js?dpl=nogit-1791235224","/_next/static/chunks/2x_dbpl3b_l2r.js?dpl=nogit-1791235224","/_next/static/chunks/13_9hjun60hj-.js?dpl=nogit-1791235224","/_next/static/chunks/2bxmml65nu49g.js?dpl=nogit-1791235224","/_next/static/chunks/0zkcxwtbsqliz.js?dpl=nogit-1791235224","/_next/static/chunks/2othipd-ghhsj.js?dpl=nogit-1791235224","/_next/static/chunks/2gjyu-jrwu-fu.js?dpl=nogit-1791235224","/_next/static/chunks/38w1x0vrld4-w.js?dpl=nogit-1791235224","/_next/static/chunks/3jpl-42mbad5n.js?dpl=nogit-1791235224","/_next/static/chunks/3kzt2piec6aoa.js?dpl=nogit-1791235224","/_next/static/chunks/19ijf68pshdcn.js?dpl=nogit-1791235224","/_next/static/chunks/2os7nxouq91eq.js?dpl=nogit-1791235224","/_next/static/chunks/3ml3vooxhiupb.js?dpl=nogit-1791235224","/_next/static/chunks/04zr-wqafvwoa.js?dpl=nogit-1791235224","/_next/static/chunks/10rmxw_yiwj40.js?dpl=nogit-1791235224","/_next/static/chunks/40lyu17d590zu.js?dpl=nogit-1791235224","/_next/static/chunks/2g-t2n9ahs3tn.js?dpl=nogit-1791235224","/_next/static/chunks/1g1suj21ijg7y.js?dpl=nogit-1791235224","/_next/static/chunks/3lz9pr5v38m4h.js?dpl=nogit-1791235224","/_next/static/chunks/2mv9l1xl_aa4j.js?dpl=nogit-1791235224","/_next/static/chunks/0v4byuex1x38h.js?dpl=nogit-1791235224","/_next/static/chunks/2oijhb0tdir9b.js?dpl=nogit-1791235224","/_next/static/chunks/2eanemqjnv2vr.js?dpl=nogit-1791235224","/_next/static/chunks/2dx-vzwyw9a3g.js?dpl=nogit-1791235224","/_next/static/chunks/0llqhg6-bxn_b.js?dpl=nogit-1791235224","/_next/static/chunks/3kxd2t9_7z5v9.js?dpl=nogit-1791235224","/_next/static/chunks/0xcrvscza9dj2.js?dpl=nogit-1791235224","/_next/static/chunks/1ku-vdy4ggo41.js?dpl=nogit-1791235224","/_next/static/chunks/1coblm1-go81_.js?dpl=nogit-1791235224","/_next/static/chunks/3hk191jki_10r.js?dpl=nogit-1791235224","/_next/static/chunks/0k-y_4mcfub-p.js?dpl=nogit-1791235224","/_next/static/chunks/42uswp7vuulbr.js?dpl=nogit-1791235224","/_next/static/chunks/2eycy5cqzjdqf.js?dpl=nogit-1791235224","/_next/static/chunks/3rgu7kqb5jfh5.js?dpl=nogit-1791235224","/_next/static/chunks/3q5gc8lbwr4b4.js?dpl=nogit-1791235224","/_next/static/chunks/3it1o0ex-l5qz.js?dpl=nogit-1791235224","/_next/static/chunks/1eyu-5752jh17.js?dpl=nogit-1791235224","/_next/static/chunks/20_c60cgo1g9g.js?dpl=nogit-1791235224","/_next/static/chunks/0wl6wuakgnlec.js?dpl=nogit-1791235224","/_next/static/chunks/03u7f9lenppb3.js?dpl=nogit-1791235224","/_next/static/chunks/0w7ow4sfu0tl_.js?dpl=nogit-1791235224","/_next/static/chunks/075i0tnxt4not.js?dpl=nogit-1791235224","/_next/static/chunks/13gj59kzq50t_.js?dpl=nogit-1791235224","/_next/static/chunks/2ryoj58czm802.js?dpl=nogit-1791235224","/_next/static/chunks/2zts3qwotnhdc.js?dpl=nogit-1791235224","/_next/static/chunks/3oiyawxg2v43g.js?dpl=nogit-1791235224","/_next/static/chunks/0a57o9qvnv-nb.js?dpl=nogit-1791235224","/_next/static/chunks/1ksyn9m68446d.js?dpl=nogit-1791235224","/_next/static/chunks/36nndwkz7iik3.js?dpl=nogit-1791235224","/_next/static/chunks/0sximxdmf77_5.js?dpl=nogit-1791235224","/_next/static/chunks/1hnuhh5t7cfaj.js?dpl=nogit-1791235224","/_next/static/chunks/3ybwgs28on6c8.js?dpl=nogit-1791235224","/_next/static/chunks/0zg1om_k0m5zv.js?dpl=nogit-1791235224","/_next/static/chunks/24srd-xd1m5z8.js?dpl=nogit-1791235224","/_next/static/chunks/30w6c7jj-mxyi.js?dpl=nogit-1791235224","/_next/static/chunks/0wv8cpdr6cvhr.js?dpl=nogit-1791235224","/_next/static/chunks/1lsxkfxy262n-.js?dpl=nogit-1791235224"],"Gingerbread"] 35:I[647935,["/_next/static/chunks/0j4ivqk4ne18h.js?dpl=nogit-1791235224","/_next/static/chunks/2_6olli-sutu3.js?dpl=nogit-1791235224","/_next/static/chunks/0zcakt1-boe_f.js?dpl=nogit-1791235224","/_next/static/chunks/0psy-n5us57a4.js?dpl=nogit-1791235224","/_next/static/chunks/3mz725-y91ph6.js?dpl=nogit-1791235224","/_next/static/chunks/2x_dbpl3b_l2r.js?dpl=nogit-1791235224","/_next/static/chunks/13_9hjun60hj-.js?dpl=nogit-1791235224","/_next/static/chunks/2bxmml65nu49g.js?dpl=nogit-1791235224","/_next/static/chunks/0zkcxwtbsqliz.js?dpl=nogit-1791235224","/_next/static/chunks/2othipd-ghhsj.js?dpl=nogit-1791235224","/_next/static/chunks/2gjyu-jrwu-fu.js?dpl=nogit-1791235224","/_next/static/chunks/38w1x0vrld4-w.js?dpl=nogit-1791235224","/_next/static/chunks/3jpl-42mbad5n.js?dpl=nogit-1791235224","/_next/static/chunks/3kzt2piec6aoa.js?dpl=nogit-1791235224","/_next/static/chunks/19ijf68pshdcn.js?dpl=nogit-1791235224","/_next/static/chunks/2os7nxouq91eq.js?dpl=nogit-1791235224","/_next/static/chunks/3ml3vooxhiupb.js?dpl=nogit-1791235224","/_next/static/chunks/04zr-wqafvwoa.js?dpl=nogit-1791235224","/_next/static/chunks/10rmxw_yiwj40.js?dpl=nogit-1791235224","/_next/static/chunks/40lyu17d590zu.js?dpl=nogit-1791235224","/_next/static/chunks/2g-t2n9ahs3tn.js?dpl=nogit-1791235224","/_next/static/chunks/1g1suj21ijg7y.js?dpl=nogit-1791235224","/_next/static/chunks/3lz9pr5v38m4h.js?dpl=nogit-1791235224","/_next/static/chunks/2mv9l1xl_aa4j.js?dpl=nogit-1791235224","/_next/static/chunks/0v4byuex1x38h.js?dpl=nogit-1791235224","/_next/static/chunks/2oijhb0tdir9b.js?dpl=nogit-1791235224","/_next/static/chunks/2eanemqjnv2vr.js?dpl=nogit-1791235224","/_next/static/chunks/2dx-vzwyw9a3g.js?dpl=nogit-1791235224","/_next/static/chunks/0llqhg6-bxn_b.js?dpl=nogit-1791235224","/_next/static/chunks/3kxd2t9_7z5v9.js?dpl=nogit-1791235224","/_next/static/chunks/0xcrvscza9dj2.js?dpl=nogit-1791235224","/_next/static/chunks/1ku-vdy4ggo41.js?dpl=nogit-1791235224","/_next/static/chunks/1coblm1-go81_.js?dpl=nogit-1791235224","/_next/static/chunks/3hk191jki_10r.js?dpl=nogit-1791235224","/_next/static/chunks/0k-y_4mcfub-p.js?dpl=nogit-1791235224","/_next/static/chunks/42uswp7vuulbr.js?dpl=nogit-1791235224","/_next/static/chunks/2eycy5cqzjdqf.js?dpl=nogit-1791235224","/_next/static/chunks/3rgu7kqb5jfh5.js?dpl=nogit-1791235224","/_next/static/chunks/3q5gc8lbwr4b4.js?dpl=nogit-1791235224","/_next/static/chunks/3it1o0ex-l5qz.js?dpl=nogit-1791235224","/_next/static/chunks/1eyu-5752jh17.js?dpl=nogit-1791235224","/_next/static/chunks/20_c60cgo1g9g.js?dpl=nogit-1791235224","/_next/static/chunks/0wl6wuakgnlec.js?dpl=nogit-1791235224","/_next/static/chunks/03u7f9lenppb3.js?dpl=nogit-1791235224","/_next/static/chunks/0w7ow4sfu0tl_.js?dpl=nogit-1791235224","/_next/static/chunks/075i0tnxt4not.js?dpl=nogit-1791235224","/_next/static/chunks/13gj59kzq50t_.js?dpl=nogit-1791235224","/_next/static/chunks/2ryoj58czm802.js?dpl=nogit-1791235224","/_next/static/chunks/2zts3qwotnhdc.js?dpl=nogit-1791235224","/_next/static/chunks/3oiyawxg2v43g.js?dpl=nogit-1791235224","/_next/static/chunks/0a57o9qvnv-nb.js?dpl=nogit-1791235224","/_next/static/chunks/1ksyn9m68446d.js?dpl=nogit-1791235224","/_next/static/chunks/36nndwkz7iik3.js?dpl=nogit-1791235224","/_next/static/chunks/0sximxdmf77_5.js?dpl=nogit-1791235224","/_next/static/chunks/1hnuhh5t7cfaj.js?dpl=nogit-1791235224","/_next/static/chunks/3ybwgs28on6c8.js?dpl=nogit-1791235224","/_next/static/chunks/0zg1om_k0m5zv.js?dpl=nogit-1791235224","/_next/static/chunks/24srd-xd1m5z8.js?dpl=nogit-1791235224","/_next/static/chunks/30w6c7jj-mxyi.js?dpl=nogit-1791235224","/_next/static/chunks/0wv8cpdr6cvhr.js?dpl=nogit-1791235224","/_next/static/chunks/1lsxkfxy262n-.js?dpl=nogit-1791235224"],"RecordResourceResume"] 36:I[830476,["/_next/static/chunks/0j4ivqk4ne18h.js?dpl=nogit-1791235224","/_next/static/chunks/2_6olli-sutu3.js?dpl=nogit-1791235224","/_next/static/chunks/0zcakt1-boe_f.js?dpl=nogit-1791235224","/_next/static/chunks/0psy-n5us57a4.js?dpl=nogit-1791235224","/_next/static/chunks/3mz725-y91ph6.js?dpl=nogit-1791235224","/_next/static/chunks/2x_dbpl3b_l2r.js?dpl=nogit-1791235224","/_next/static/chunks/13_9hjun60hj-.js?dpl=nogit-1791235224","/_next/static/chunks/2bxmml65nu49g.js?dpl=nogit-1791235224","/_next/static/chunks/0zkcxwtbsqliz.js?dpl=nogit-1791235224","/_next/static/chunks/2othipd-ghhsj.js?dpl=nogit-1791235224","/_next/static/chunks/2gjyu-jrwu-fu.js?dpl=nogit-1791235224","/_next/static/chunks/38w1x0vrld4-w.js?dpl=nogit-1791235224","/_next/static/chunks/3jpl-42mbad5n.js?dpl=nogit-1791235224","/_next/static/chunks/3kzt2piec6aoa.js?dpl=nogit-1791235224","/_next/static/chunks/19ijf68pshdcn.js?dpl=nogit-1791235224","/_next/static/chunks/2os7nxouq91eq.js?dpl=nogit-1791235224","/_next/static/chunks/3ml3vooxhiupb.js?dpl=nogit-1791235224","/_next/static/chunks/04zr-wqafvwoa.js?dpl=nogit-1791235224","/_next/static/chunks/10rmxw_yiwj40.js?dpl=nogit-1791235224","/_next/static/chunks/40lyu17d590zu.js?dpl=nogit-1791235224","/_next/static/chunks/2g-t2n9ahs3tn.js?dpl=nogit-1791235224","/_next/static/chunks/1g1suj21ijg7y.js?dpl=nogit-1791235224","/_next/static/chunks/3lz9pr5v38m4h.js?dpl=nogit-1791235224","/_next/static/chunks/2mv9l1xl_aa4j.js?dpl=nogit-1791235224","/_next/static/chunks/0v4byuex1x38h.js?dpl=nogit-1791235224","/_next/static/chunks/2oijhb0tdir9b.js?dpl=nogit-1791235224","/_next/static/chunks/2eanemqjnv2vr.js?dpl=nogit-1791235224","/_next/static/chunks/2dx-vzwyw9a3g.js?dpl=nogit-1791235224","/_next/static/chunks/0llqhg6-bxn_b.js?dpl=nogit-1791235224","/_next/static/chunks/3kxd2t9_7z5v9.js?dpl=nogit-1791235224","/_next/static/chunks/0xcrvscza9dj2.js?dpl=nogit-1791235224","/_next/static/chunks/1ku-vdy4ggo41.js?dpl=nogit-1791235224","/_next/static/chunks/1coblm1-go81_.js?dpl=nogit-1791235224","/_next/static/chunks/3hk191jki_10r.js?dpl=nogit-1791235224","/_next/static/chunks/0k-y_4mcfub-p.js?dpl=nogit-1791235224","/_next/static/chunks/42uswp7vuulbr.js?dpl=nogit-1791235224","/_next/static/chunks/2eycy5cqzjdqf.js?dpl=nogit-1791235224","/_next/static/chunks/3rgu7kqb5jfh5.js?dpl=nogit-1791235224","/_next/static/chunks/3q5gc8lbwr4b4.js?dpl=nogit-1791235224","/_next/static/chunks/3it1o0ex-l5qz.js?dpl=nogit-1791235224","/_next/static/chunks/1eyu-5752jh17.js?dpl=nogit-1791235224","/_next/static/chunks/20_c60cgo1g9g.js?dpl=nogit-1791235224","/_next/static/chunks/0wl6wuakgnlec.js?dpl=nogit-1791235224","/_next/static/chunks/03u7f9lenppb3.js?dpl=nogit-1791235224","/_next/static/chunks/0w7ow4sfu0tl_.js?dpl=nogit-1791235224","/_next/static/chunks/075i0tnxt4not.js?dpl=nogit-1791235224","/_next/static/chunks/13gj59kzq50t_.js?dpl=nogit-1791235224","/_next/static/chunks/2ryoj58czm802.js?dpl=nogit-1791235224","/_next/static/chunks/2zts3qwotnhdc.js?dpl=nogit-1791235224","/_next/static/chunks/3oiyawxg2v43g.js?dpl=nogit-1791235224","/_next/static/chunks/0a57o9qvnv-nb.js?dpl=nogit-1791235224","/_next/static/chunks/1ksyn9m68446d.js?dpl=nogit-1791235224","/_next/static/chunks/36nndwkz7iik3.js?dpl=nogit-1791235224","/_next/static/chunks/0sximxdmf77_5.js?dpl=nogit-1791235224","/_next/static/chunks/1hnuhh5t7cfaj.js?dpl=nogit-1791235224","/_next/static/chunks/3ybwgs28on6c8.js?dpl=nogit-1791235224","/_next/static/chunks/0zg1om_k0m5zv.js?dpl=nogit-1791235224","/_next/static/chunks/24srd-xd1m5z8.js?dpl=nogit-1791235224","/_next/static/chunks/30w6c7jj-mxyi.js?dpl=nogit-1791235224","/_next/static/chunks/0wv8cpdr6cvhr.js?dpl=nogit-1791235224","/_next/static/chunks/1lsxkfxy262n-.js?dpl=nogit-1791235224"],"FlashcardAccessBranch"] 37:I[315438,["/_next/static/chunks/0j4ivqk4ne18h.js?dpl=nogit-1791235224","/_next/static/chunks/2_6olli-sutu3.js?dpl=nogit-1791235224","/_next/static/chunks/0zcakt1-boe_f.js?dpl=nogit-1791235224","/_next/static/chunks/0psy-n5us57a4.js?dpl=nogit-1791235224","/_next/static/chunks/3mz725-y91ph6.js?dpl=nogit-1791235224","/_next/static/chunks/2x_dbpl3b_l2r.js?dpl=nogit-1791235224","/_next/static/chunks/13_9hjun60hj-.js?dpl=nogit-1791235224","/_next/static/chunks/2bxmml65nu49g.js?dpl=nogit-1791235224","/_next/static/chunks/0zkcxwtbsqliz.js?dpl=nogit-1791235224","/_next/static/chunks/2othipd-ghhsj.js?dpl=nogit-1791235224","/_next/static/chunks/2gjyu-jrwu-fu.js?dpl=nogit-1791235224","/_next/static/chunks/38w1x0vrld4-w.js?dpl=nogit-1791235224","/_next/static/chunks/3jpl-42mbad5n.js?dpl=nogit-1791235224","/_next/static/chunks/3kzt2piec6aoa.js?dpl=nogit-1791235224","/_next/static/chunks/19ijf68pshdcn.js?dpl=nogit-1791235224","/_next/static/chunks/2os7nxouq91eq.js?dpl=nogit-1791235224","/_next/static/chunks/3ml3vooxhiupb.js?dpl=nogit-1791235224","/_next/static/chunks/04zr-wqafvwoa.js?dpl=nogit-1791235224","/_next/static/chunks/10rmxw_yiwj40.js?dpl=nogit-1791235224","/_next/static/chunks/40lyu17d590zu.js?dpl=nogit-1791235224","/_next/static/chunks/2g-t2n9ahs3tn.js?dpl=nogit-1791235224","/_next/static/chunks/1g1suj21ijg7y.js?dpl=nogit-1791235224","/_next/static/chunks/3lz9pr5v38m4h.js?dpl=nogit-1791235224","/_next/static/chunks/2mv9l1xl_aa4j.js?dpl=nogit-1791235224","/_next/static/chunks/0v4byuex1x38h.js?dpl=nogit-1791235224","/_next/static/chunks/2oijhb0tdir9b.js?dpl=nogit-1791235224","/_next/static/chunks/2eanemqjnv2vr.js?dpl=nogit-1791235224","/_next/static/chunks/2dx-vzwyw9a3g.js?dpl=nogit-1791235224","/_next/static/chunks/0llqhg6-bxn_b.js?dpl=nogit-1791235224","/_next/static/chunks/3kxd2t9_7z5v9.js?dpl=nogit-1791235224","/_next/static/chunks/0xcrvscza9dj2.js?dpl=nogit-1791235224","/_next/static/chunks/1ku-vdy4ggo41.js?dpl=nogit-1791235224","/_next/static/chunks/1coblm1-go81_.js?dpl=nogit-1791235224","/_next/static/chunks/3hk191jki_10r.js?dpl=nogit-1791235224","/_next/static/chunks/0k-y_4mcfub-p.js?dpl=nogit-1791235224","/_next/static/chunks/42uswp7vuulbr.js?dpl=nogit-1791235224","/_next/static/chunks/2eycy5cqzjdqf.js?dpl=nogit-1791235224","/_next/static/chunks/3rgu7kqb5jfh5.js?dpl=nogit-1791235224","/_next/static/chunks/3q5gc8lbwr4b4.js?dpl=nogit-1791235224","/_next/static/chunks/3it1o0ex-l5qz.js?dpl=nogit-1791235224","/_next/static/chunks/1eyu-5752jh17.js?dpl=nogit-1791235224","/_next/static/chunks/20_c60cgo1g9g.js?dpl=nogit-1791235224","/_next/static/chunks/0wl6wuakgnlec.js?dpl=nogit-1791235224","/_next/static/chunks/03u7f9lenppb3.js?dpl=nogit-1791235224","/_next/static/chunks/0w7ow4sfu0tl_.js?dpl=nogit-1791235224","/_next/static/chunks/075i0tnxt4not.js?dpl=nogit-1791235224","/_next/static/chunks/13gj59kzq50t_.js?dpl=nogit-1791235224","/_next/static/chunks/2ryoj58czm802.js?dpl=nogit-1791235224","/_next/static/chunks/2zts3qwotnhdc.js?dpl=nogit-1791235224","/_next/static/chunks/3oiyawxg2v43g.js?dpl=nogit-1791235224","/_next/static/chunks/0a57o9qvnv-nb.js?dpl=nogit-1791235224","/_next/static/chunks/1ksyn9m68446d.js?dpl=nogit-1791235224","/_next/static/chunks/36nndwkz7iik3.js?dpl=nogit-1791235224","/_next/static/chunks/0sximxdmf77_5.js?dpl=nogit-1791235224","/_next/static/chunks/1hnuhh5t7cfaj.js?dpl=nogit-1791235224","/_next/static/chunks/3ybwgs28on6c8.js?dpl=nogit-1791235224","/_next/static/chunks/0zg1om_k0m5zv.js?dpl=nogit-1791235224","/_next/static/chunks/24srd-xd1m5z8.js?dpl=nogit-1791235224","/_next/static/chunks/30w6c7jj-mxyi.js?dpl=nogit-1791235224","/_next/static/chunks/0wv8cpdr6cvhr.js?dpl=nogit-1791235224","/_next/static/chunks/1lsxkfxy262n-.js?dpl=nogit-1791235224"],"TopicFlashcardRunner"] 24:[["$","$L33",null,{"feature":"flashcards"}],"$L34",["$","$L35",null,{"href":"/igcse/maths/edexcel/grade-6-6-13-similar-shapes-area-and-volume/flashcards","label":"Similar Shapes (Area and Volume)","subjectCode":"igcse_maths","surface":"flashcards"}],["$","$L36",null,{"runner":["$","div",null,{"className":"min-h-0 flex-1 overflow-hidden bg-background","children":["$","$L37",null,{"decks":[{"id":"0e2f6cea-a5dc-4da6-9a3d-9dd7cbaa4617","title":"Similar Shapes (Area and Volume) Flashcards","topicId":"c8e2fc4b-279e-481c-939f-f524d1e492d0","topicSlug":"grade-6-6-13-similar-shapes-area-and-volume","cards":[{"id":"ac941755-a191-4d37-a8cd-be6e28120a6a","cardType":"basic","points":2,"front":"What defines two 3D shapes as mathematically similar?","back":"

Identical shape
Matching lengths enlarged/reduced by the same multiplier

","mcqOptions":[{"text":"

Identical shape
Matching lengths enlarged/reduced by the same multiplier

","correct":true,"explanation":"Similar 3D shapes have the same form, and every pair of corresponding lengths has one common ratio. This shared multiplier is the linear scale factor."},{"text":"

Identical shape
Matching lengths enlarged/reduced by the same amount

","correct":false,"explanation":"Adding the same amount to each length does not preserve proportions and can change the shape. Corresponding lengths must instead be multiplied by the same scale factor."},{"text":"

Identical volume
Matching lengths enlarged/reduced by a different multiplier

","correct":false,"explanation":"Equal volumes do not guarantee identical shape, and different length multipliers distort proportions. Similar shapes require identical shape with every corresponding length changed by the same multiplier."},{"text":"

Identical surface area
Matching lengths enlarged/reduced by the same amount

","correct":false,"explanation":"Equal surface areas can occur for shapes that are not similar, while equal additions do not preserve proportions. Similarity requires identical shape and one common length multiplier."}]},{"id":"b739a5f6-31c6-4898-b7ad-76f2481ded02","cardType":"cloze","points":null,"front":"If the linear scale factor is $k$, then the area scale factor is $\\text{[...]}$.","back":"If the linear scale factor is $k$, then the area scale factor is $\\boxed{k^2}$.","mcqOptions":[{"text":"$$k^2$","correct":true,"explanation":"Area is two-dimensional, so both corresponding length dimensions are multiplied by $k$. Their combined effect gives an area scale factor of $k \\times k = k^2$."},{"text":"$$k$","correct":false,"explanation":"Using the linear scale factor unchanged ignores that area has two dimensions. Each dimension scales by $k$, so the correct area scale factor is $k^2$."},{"text":"$$2k$","correct":false,"explanation":"Doubling the scale factor confuses repeated multiplication with addition. Two scaled dimensions give $k \\times k$, so the correct area scale factor is $k^2$."},{"text":"$$k^3$","correct":false,"explanation":"Cubing the linear scale factor applies to volume because volume is three-dimensional. Area is two-dimensional, so its scale factor is $k^2$."}]},{"id":"32ca73b4-0121-4a28-92a2-951602a3afe0","cardType":"cloze","points":null,"front":"If the linear scale factor is $k$, then the volume scale factor is $\\text{[...]}$.","back":"If the linear scale factor is $k$, then the volume scale factor is $\\boxed{k^3}$.","mcqOptions":[{"text":"$$k^3$","correct":true,"explanation":"Volume involves three dimensions, each multiplied by $k$. Multiplying their effects gives $k \\times k \\times k = k^3$ as the volume scale factor."},{"text":"$$k^2$","correct":false,"explanation":"Squaring the scale factor accounts for only two dimensions and therefore applies to area. Volume has three dimensions, so the correct scale factor is $k^3$."},{"text":"$$3k$","correct":false,"explanation":"Multiplying $k$ by 3 confuses repeated multiplication with addition. Three scaled dimensions produce $k \\times k \\times k$, giving a volume scale factor of $k^3$."},{"text":"$$k^4$","correct":false,"explanation":"Raising $k$ to the fourth power incorrectly introduces an extra dimension. A volume has three dimensions, so its scale factor is $k^3$."}]},{"id":"0f6b78b0-e3d5-4424-96e0-7a84ef02d282","cardType":"basic","points":1,"front":"If the radius of a shape increases by a factor of 3, what is the area scale factor?","back":"$$9$ (since $3^2 = 9$)","mcqOptions":[{"text":"$$9$ (since $3^2 = 9$)","correct":true,"explanation":"Area scales with the square of the linear scale factor. A radius multiplier of 3 therefore gives an area scale factor of $3^2 = 9$."},{"text":"$$3$ (since $3^1 = 3$)","correct":false,"explanation":"Using the radius multiplier directly treats area as one-dimensional. Area scales with the square of the linear factor, so the correct scale factor is $3^2 = 9$."},{"text":"$$6$ (since $3 \\times 2 = 6$)","correct":false,"explanation":"Multiplying the linear factor by the number of dimensions is not the area rule. The factor must be squared, giving $3^2 = 9$."},{"text":"$$27$ (since $3^3 = 27$)","correct":false,"explanation":"Cubing the linear scale factor gives the volume scale factor, not the area scale factor. Area uses the square, so the correct value is $3^2 = 9$."}]},{"id":"9b40f67e-17e3-419f-8110-016edb02d81d","cardType":"basic","points":1,"front":"If the height of a shape increases by a factor of 2, what is the volume scale factor?","back":"$$8$ (since $2^3 = 8$)","mcqOptions":[{"text":"$$8$ (since $2^3 = 8$)","correct":true,"explanation":"Volume is three-dimensional, so a linear scale factor of 2 is applied three times. The volume scale factor is therefore $2^3 = 8$."},{"text":"$$2$ (since $2^1 = 2$)","correct":false,"explanation":"Using the height multiplier directly treats volume as one-dimensional. For similar shapes, all three dimensions scale by 2, so the volume scale factor is $2^3 = 8$."},{"text":"$$4$ (since $2^2 = 4$)","correct":false,"explanation":"Squaring the linear scale factor gives the area scale factor, not the volume scale factor. Volume is three-dimensional, so the correct calculation is $2^3 = 8$."},{"text":"$$6$ (since $2 \\times 3 = 6$)","correct":false,"explanation":"Multiplying the scale factor by three confuses repeated multiplication with addition. Volume requires $2 \\times 2 \\times 2$, producing the correct scale factor of 8."}]},{"id":"fea2eff5-f57e-4e4e-b7de-95bc6cc961fe","cardType":"basic","points":1,"front":"How is the linear scale factor ($k$) calculated from the area scale factor?","back":"$$k = \\sqrt{\\text{area scale factor}}$","mcqOptions":[{"text":"$$k = \\sqrt{\\text{area scale factor}}$","correct":true,"explanation":"The area scale factor equals $k^2$, so reversing the relationship requires taking its positive square root. This gives the linear scale factor $k$."},{"text":"$$k = \\left(\\text{area scale factor}\\right)^2$","correct":false,"explanation":"Squaring moves from a linear scale factor to an area scale factor rather than reversing the process. The linear scale factor is the square root of the area scale factor."},{"text":"$$k = \\sqrt[3]{\\text{area scale factor}}$","correct":false,"explanation":"A cube root reverses a volume scale factor because volume scales with $k^3$. Area scales with $k^2$, so its square root gives $k$."},{"text":"$$k = \\dfrac{\\text{area scale factor}}{2}$","correct":false,"explanation":"Dividing by 2 treats the exponent as an ordinary multiplier. Since area scale factor equals $k^2$, the inverse operation is a square root, not division."}]},{"id":"ed5c4464-afcb-428f-8313-1f358bef00d1","cardType":"basic","points":1,"front":"How is the linear scale factor ($k$) calculated from the volume scale factor?","back":"$$k = \\sqrt[3]{\\text{volume scale factor}}$","mcqOptions":[{"text":"$$k = \\sqrt[3]{\\text{volume scale factor}}$","correct":true,"explanation":"The volume scale factor equals $k^3$, so taking its cube root reverses the cubing and recovers the linear scale factor $k$."},{"text":"$$k = \\sqrt{\\text{volume scale factor}}$","correct":false,"explanation":"A square root reverses an area scale factor, which equals $k^2$. Volume scales with $k^3$, so the linear scale factor requires a cube root."},{"text":"$$k = \\left(\\text{volume scale factor}\\right)^3$","correct":false,"explanation":"Cubing would increase the volume scale factor rather than recover the original length ratio. Since volume scale factor equals $k^3$, its cube root gives $k$."},{"text":"$$k = \\dfrac{\\text{volume scale factor}}{3}$","correct":false,"explanation":"Dividing by 3 incorrectly treats the power as a multiplier. To reverse $k^3$, take the cube root of the volume scale factor."}]},{"id":"cd0f6865-b0ca-4db5-8d90-27ea3b7484be","cardType":"cloze","points":null,"front":"Surface area changes by the square of the scale factor because it is a [ ] measurement, whereas volume uses the cube because it is 3D.","back":"Surface area changes by the square of the scale factor because it is a 2D measurement, whereas volume uses the cube because it is 3D.","mcqOptions":[{"text":"2D","correct":true,"explanation":"Surface area measures a two-dimensional covering, so two scaled length dimensions are multiplied together. This is why its scale factor is the square of the linear factor."},{"text":"1D","correct":false,"explanation":"A one-dimensional measurement describes length rather than surface area. Surface area extends in two dimensions, so it scales with the square of the linear scale factor."},{"text":"3D","correct":false,"explanation":"Three-dimensional scaling applies to volume because volume involves length, width and height. Surface area is a two-dimensional measurement, so it uses the square of the scale factor."},{"text":"4D","correct":false,"explanation":"Four-dimensional measurement is not involved in ordinary similar solids. Surface area has two dimensions, so its scale factor is $k^2$, while volume uses $k^3$."}]},{"id":"b1de99ba-336f-46ff-b524-d76015810a3a","cardType":"cloze","points":null,"front":"Surface area changes by the square of the scale factor because it is a 2D measurement, whereas volume uses the cube because it is [ ].","back":"Surface area changes by the square of the scale factor because it is a 2D measurement, whereas volume uses the cube because it is 3D.","mcqOptions":[{"text":"3D","correct":true,"explanation":"Volume measures three dimensions - length, width and height - so a linear scale factor of $k$ produces a volume scale factor of $k^3$."},{"text":"2D","correct":false,"explanation":"Surface area is 2D, but volume is not. Volume measures three dimensions, so its scale factor is the cube of the linear scale factor."},{"text":"1D","correct":false,"explanation":"A 1D measurement is a length and therefore uses the linear scale factor $k$. Volume is 3D, which is why it uses $k^3$."},{"text":"4D","correct":false,"explanation":"Ordinary geometric solids have three spatial dimensions, not four. Volume is 3D, so corresponding volumes change according to the cube of the linear scale factor."}]},{"id":"36d78405-4a39-402d-b300-e9f2d9399063","cardType":"basic","points":1,"front":"When moving from a larger similar shape to a smaller similar shape, how do you use the scale factor?","back":"Divide by the scale factor (e.g. divide by $k$ for length, $k^2$ for area).","mcqOptions":[{"text":"Divide by the scale factor (e.g. divide by $k$ for length, $k^2$ for area).","correct":true,"explanation":"Moving from a larger shape to a smaller similar shape reverses the enlargement, so divide by the relevant length, area or volume scale factor."},{"text":"Multiply by the scale factor (e.g. multiply by $k$ for length, $k^2$ for area).","correct":false,"explanation":"Multiplication applies when moving from the smaller shape to the larger one. Moving from larger to smaller requires division by $k$, $k^2$ or $k^3$ as appropriate."},{"text":"Subtract the scale factor (e.g. subtract $k$ for length, $k^2$ for area).","correct":false,"explanation":"Subtracting a scale factor confuses multiplicative scaling with an additive change. Similar dimensions are found by dividing by $k$, $k^2$ or $k^3$ when moving to the smaller shape."},{"text":"Divide by the reciprocal scale factor (e.g. divide by $\\frac{1}{k}$ for length, $\\frac{1}{k^2}$ for area).","correct":false,"explanation":"Taking a reciprocal and then dividing is equivalent to multiplying by the original factor, so the measurement grows. Instead, divide directly by the relevant scale factor."}]},{"id":"401f6cf1-d33a-4085-a20d-5141f4e06522","cardType":"basic","points":3,"front":"What is the 3-step process to find a missing volume if you are only given two matching surface areas?","back":"

Find the area scale factor
Square root it to find the linear scale factor ($k$)
Cube $k$ to find the volume scale factor ($k^3$)

","mcqOptions":[{"text":"

Find the area scale factor
Square root it to find the linear scale factor ($k$)
Cube $k$ to find the volume scale factor ($k^3$)

","correct":true,"explanation":"The surface-area ratio equals $k^2$, so taking its square root gives $k$. Cubing that linear scale factor then gives the corresponding volume ratio $k^3$."},{"text":"

Find the area scale factor
Square it to find the linear scale factor ($k$)
Cube $k$ to find the volume scale factor ($k^3$)

","correct":false,"explanation":"The area ratio is $k^2$, so it must be square-rooted rather than squared to recover $k$. The required volume scale factor is then $k^3$."},{"text":"

Find the area scale factor
Square root it to find the linear scale factor ($k$)
Multiply $k$ by $3$ to find the volume scale factor ($3k$)

","correct":false,"explanation":"Multiplying the area ratio by three does not convert it into a volume ratio. First square-root the area scale factor to obtain $k$, then cube $k$."},{"text":"

Find the area scale factor
Cube root it to find the linear scale factor ($k$)
Square $k$ to find the volume scale factor ($k^2$)

","correct":false,"explanation":"A cube root is used to recover a linear factor from a volume ratio, not an area ratio. Square-root the area scale factor, then cube the resulting $k$."}]},{"id":"47b4656d-d182-4878-ba12-18dc0976e6cd","cardType":"basic","points":1,"front":"What is a frustum?","back":"The solid remaining when the top of a cone or pyramid is cut off parallel to the base.","mcqOptions":[{"text":"The solid remaining when the top of a cone or pyramid is cut off parallel to the base.","correct":true,"explanation":"A cut parallel to the base removes a smaller similar cone or pyramid. The portion left between the parallel faces is called a frustum."},{"text":"The solid remaining when the top of a cone or pyramid is cut off perpendicular to the base.","correct":false,"explanation":"A perpendicular cut does not generally produce the standard smaller similar solid or parallel circular faces. A frustum is formed by cutting parallel to the base."},{"text":"The solid remaining when the base of a cone or pyramid is cut off parallel to the top.","correct":false,"explanation":"Removing the bottom is not the standard construction described for a frustum. It is the solid left when the top is cut off parallel to the base."},{"text":"The solid remaining when the top of a sphere or hemisphere is cut off parallel to the base.","correct":false,"explanation":"A sphere has no flat base to which this cut can be parallel. A frustum comes from cutting the top off a cone or pyramid parallel to its base."}]},{"id":"a5a88ad0-4a6f-48da-9cda-60c5fa9e9452","cardType":"basic","points":1,"front":"How is the volume of a frustum calculated using similar shapes?","back":"$$\\text{Large cone volume} - \\text{Small cone volume}$","mcqOptions":[{"text":"$$\\text{Large cone volume} - \\text{Small cone volume}$","correct":true,"explanation":"The large cone can be viewed as the frustum plus the smaller similar cone removed from its top, so subtracting their volumes leaves the frustum."},{"text":"$$\\text{Large cone volume} + \\text{Small cone volume}$","correct":false,"explanation":"Adding counts the removed small cone as part of the required solid. The frustum is what remains, so subtract the small cone volume from the large cone volume."},{"text":"$$\\text{Small cone volume} - \\text{Large cone volume}$","correct":false,"explanation":"Reversing the subtraction gives a negative value because the small cone has less volume. The correct order is large cone volume minus small cone volume."},{"text":"$$\\text{Large cone volume} \\div \\text{Small cone volume}$","correct":false,"explanation":"Dividing the two volumes gives a ratio rather than the volume of the remaining solid. The frustum’s volume is found by subtracting the smaller volume from the larger."}]},{"id":"7f35d84c-04f2-49d0-b4c3-69fa7da2913c","cardType":"basic","points":1,"front":"What is the formula for the volume of a cone?","back":"$$V = \\frac{1}{3}\\pi r^2h$","mcqOptions":[{"text":"$$V = \\frac{1}{3}\\pi r^2h$","correct":true,"explanation":"A cone has one-third the volume of a cylinder with the same base radius and perpendicular height, giving $V=\\frac{1}{3}\\pi r^2h$."},{"text":"$$V = \\pi r^2h$","correct":false,"explanation":"Omitting the factor of $\\frac{1}{3}$ gives the volume of the corresponding cylinder. A cone occupies one-third of that cylinder’s volume."},{"text":"$$V = \\frac{1}{3}2\\pi rh$","correct":false,"explanation":"The base is a circle with area $\\pi r^2$, not $2\\pi r$. Multiply that area by the height and by $\\frac{1}{3}$."},{"text":"$$V = \\frac{1}{3}\\pi rh^2$","correct":false,"explanation":"Squaring the height instead of the radius uses the wrong dimensions for the circular base. The radius must be squared, giving $V=\\frac{1}{3}\\pi r^2h$."}]},{"id":"7dca68f0-f9f7-4892-937e-c6b7e810c1f7","cardType":"cloze","points":null,"front":"Before using the cone volume formula, you must check if you are given the [ ]; if so, you must halve it to find the radius.","back":"Before using the cone volume formula, you must check if you are given the diameter; if so, you must halve it to find the radius.","mcqOptions":[{"text":"diameter","correct":true,"explanation":"The cone formula uses the radius, while diagrams may supply the full width through the centre. That measurement is the diameter and must be halved."},{"text":"radius","correct":false,"explanation":"The radius is already the required distance from the centre to the circumference, so it should not be halved. The measurement requiring halving is the diameter."},{"text":"height","correct":false,"explanation":"The perpendicular height is used directly in the cone volume formula and is unrelated to finding the radius. It is the diameter that must be halved."},{"text":"slant height","correct":false,"explanation":"The slant height runs along the cone’s surface and is not halved to obtain the radius. The diameter is the full width that must be halved."}]},{"id":"6b7086d0-c5b2-4e8c-9a02-1ec5293c34f2","cardType":"cloze","points":null,"front":"Before using the cone volume formula, you must check if you are given the diameter; if so, you must [ ] it to find the radius.","back":"Before using the cone volume formula, you must check if you are given the diameter; if so, you must halve it to find the radius.","mcqOptions":[{"text":"halve","correct":true,"explanation":"The diameter spans two radii, so dividing it by two gives the radius required in $V=\\frac{1}{3}\\pi r^2h$."},{"text":"double","correct":false,"explanation":"Doubling the diameter makes the value four times the true radius, leading to a severely inflated value after squaring. The diameter must instead be halved."},{"text":"square","correct":false,"explanation":"Squaring the diameter does not convert it to a radius and also creates squared units. Divide the diameter by two, or halve it, to obtain the radius."},{"text":"square root","correct":false,"explanation":"Square-rooting has no role in converting diameter to radius. Since a diameter consists of two radii, it must be halved before substitution."}]},{"id":"2ae8defe-4ae4-4887-83bd-54440b56c7ba","cardType":"basic","points":2,"front":"Two similar cones have surface areas of $50\\text{ cm}^2$ and $200\\text{ cm}^2$. What is the linear scale factor?","back":"

Area scale factor $= 4$
$k = \\sqrt{4} = 2$

","mcqOptions":[{"text":"

Area scale factor $= 4$
$k = \\sqrt{4} = 2$

","correct":true,"explanation":"The area scale factor is $200 \\div 50 = 4$. For similar shapes, area scales with $k^2$, so the linear scale factor is $\\sqrt{4}=2$."},{"text":"

Area scale factor $= 4$
$k = 4$

","correct":false,"explanation":"Using the area scale factor directly as the linear scale factor ignores that areas scale with $k^2$. Since $k^2=4$, the correct linear scale factor is $k=2$."},{"text":"

Area scale factor $= 2$
$k = \\sqrt{2}$

","correct":false,"explanation":"Dividing $200$ by $50$ gives $4$, not $2$, so the area scale factor has been calculated incorrectly. Taking its square root gives the correct linear scale factor, $k=2$."},{"text":"

Area scale factor $= 4$
$k = 4^2 = 16$

","correct":false,"explanation":"Squaring the area scale factor reverses the required operation. Because area scales as $k^2$, take the square root of $4$ to obtain the linear scale factor $k=2$."}]},{"id":"7976ce75-1487-4193-82a2-498503715504","cardType":"basic","points":1,"front":"If the volume scale factor is 27, by what factor does a length on the shape change?","back":"$$3$ (since $\\sqrt[3]{27} = 3$)","mcqOptions":[{"text":"$$3$ (since $\\sqrt[3]{27} = 3$)","correct":true,"explanation":"For similar solids, the volume scale factor is the cube of the linear scale factor. Therefore, the length factor is $\\sqrt[3]{27}=3$."},{"text":"$$9$ (since $27^{2/3} = 9$)","correct":false,"explanation":"Raising $27$ to the power $2/3$ gives the area scale factor, not the length factor. Lengths change by $\\sqrt[3]{27}=3$."},{"text":"$$27$ (using the volume factor directly)","correct":false,"explanation":"Using $27$ directly confuses the volume scale factor with the linear scale factor. Since volume scales with the cube of length, the length factor is $\\sqrt[3]{27}=3$."},{"text":"$$\\sqrt{27}$ (using the square root)","correct":false,"explanation":"Taking a square root is appropriate when converting an area scale factor to a length factor. For volume, take the cube root, giving $\\sqrt[3]{27}=3$."}]},{"id":"81e46118-be71-42cc-bff6-c65d93a90f2c","cardType":"basic","points":1,"front":"If the ratio of lengths in two similar solids is $2:5$, what is the ratio of their surface areas?","back":"$$4:25$ (since $2^2:5^2$)","mcqOptions":[{"text":"$$4:25$ (since $2^2:5^2$)","correct":true,"explanation":"Surface areas of similar solids scale with the square of the length ratio. Squaring both parts of $2:5$ gives $2^2:5^2=4:25$."},{"text":"$$2:5$ (using the length ratio directly)","correct":false,"explanation":"Using $2:5$ directly overlooks that surface area is two-dimensional. Squaring each part of the length ratio gives the correct surface-area ratio, $4:25$."},{"text":"$$8:125$ (since $2^3:5^3$)","correct":false,"explanation":"Cubing the length ratio gives the volume ratio, not the surface-area ratio. Surface area requires squaring, so $2^2:5^2=4:25$."},{"text":"$$4:10$ (since $2^2:2\\times5$)","correct":false,"explanation":"Only the first part of the ratio has been squared correctly. Both corresponding lengths must be squared, producing $2^2:5^2=4:25$."}]},{"id":"0d63c594-ea5d-4041-991d-d4449facee5f","cardType":"basic","points":1,"front":"If the ratio of lengths in two similar solids is $2:5$, what is the ratio of their volumes?","back":"$$8:125$ (since $2^3:5^3$)","mcqOptions":[{"text":"$$8:125$ (since $2^3:5^3$)","correct":true,"explanation":"Volumes of similar solids scale with the cube of the length ratio. Cubing both parts of $2:5$ gives $2^3:5^3=8:125$."},{"text":"$$4:25$ (since $2^2:5^2$)","correct":false,"explanation":"Squaring the length ratio gives the surface-area ratio, not the volume ratio. Volume is three-dimensional, so cubing gives $2^3:5^3=8:125$."},{"text":"$$2:5$ (using the length ratio directly)","correct":false,"explanation":"Using the length ratio directly ignores that volume changes in three dimensions. Cube both parts of $2:5$ to obtain the correct volume ratio, $8:125$."},{"text":"$$8:25$ (since $2^3:5^2$)","correct":false,"explanation":"The numerator has been cubed but the denominator has only been squared. Both parts of the length ratio must be cubed, giving $8:125$."}]},{"id":"532aab89-f379-450b-8bb5-5f471e306ed6","cardType":"cloze","points":null,"front":"If the surface areas of two similar solids are in the ratio $25:64$, their length ratio is $\\text{[...]}$.","back":"If the surface areas of two similar solids are in the ratio $25:64$, their length ratio is $\\boxed{5:8}$.","mcqOptions":[{"text":"$$5:8$","correct":true,"explanation":"The length ratio is the square root of the surface-area ratio. Taking square roots gives $\\sqrt{25}:\\sqrt{64}=5:8$."},{"text":"$$25:64$","correct":false,"explanation":"This keeps the surface-area ratio unchanged, confusing it with the length ratio. Taking the square root of each part gives the correct length ratio, $5:8$."},{"text":"$$8:5$","correct":false,"explanation":"The square roots are correct but their order has been reversed. Keeping the original correspondence, $\\sqrt{25}:\\sqrt{64}$ gives the length ratio $5:8$."},{"text":"$$625:4096$","correct":false,"explanation":"Squaring $25:64$ moves from area towards a fourth-power relationship rather than back to length. Taking square roots instead gives the correct ratio $5:8$."}]},{"id":"90f2931d-b471-47a3-a88e-4ba25a85c9fa","cardType":"basic","points":1,"front":"When finding the volume of a frustum, why are the large cone and the removed top cone considered mathematically similar?","back":"Because the top is cut parallel to the base, preserving the same shape and angles.","mcqOptions":[{"text":"Because the top is cut parallel to the base, preserving the same shape and angles.","correct":true,"explanation":"A cut parallel to the cone’s base creates a smaller cone with equal corresponding angles and proportional lengths. Therefore, the removed cone and original large cone are mathematically similar."},{"text":"Because the top is cut perpendicular to the base, preserving the same shape and angles.","correct":false,"explanation":"A perpendicular cut would pass vertically through the cone and would not form a smaller similar cone. The cut must be parallel to the base to preserve corresponding angles and proportions."},{"text":"Because the top is cut through the cone’s midpoint, preserving the same shape and angles.","correct":false,"explanation":"Passing through a midpoint does not by itself guarantee similarity or preserve the circular cross-section appropriately. Similarity follows specifically because the cutting plane is parallel to the cone’s base."},{"text":"Because the top is cut at the same radius as the base, preserving the same shape and angles.","correct":false,"explanation":"A removed top cone normally has a smaller radius than the original cone, so equal radii are not required. Similarity is guaranteed by making the cut parallel to the base."}]}]}],"examCountdown":{"examDate":"2027-05-14","paperLabel":"Paper 1"},"heading":"Similar Shapes (Area and Volume)","title":"Similar Shapes (Area and Volume) Flashcards","topbar":{"backHref":"/igcse/maths/edexcel/flashcards"},"topicSidebar":"$L38"}]}],"topicId":"c8e2fc4b-279e-481c-939f-f524d1e492d0","walled":"$L39"}],"$L3a","$L3b"]