{"id":1584,"date":"2020-08-31T19:35:45","date_gmt":"2020-08-31T17:35:45","guid":{"rendered":"http:\/\/matma.com.pl\/?p=1584"},"modified":"2020-08-31T19:36:59","modified_gmt":"2020-08-31T17:36:59","slug":"bryly-2","status":"publish","type":"post","link":"http:\/\/matma.com.pl\/?p=1584","title":{"rendered":"Graniastos\u0142upy."},"content":{"rendered":"<p><span style=\"font-family: 'book antiqua', palatino, serif;\"><span style=\"color: #4472c4;\"><strong>Graniastos\u0142up<\/strong><\/span>\u00a0to bry\u0142a przestrzenna, kt\u00f3rego podstaw\u0105 jest wielok\u0105t, a \u015bciany boczne s\u0105 r\u00f3wnoleg\u0142obokami. Obie podstawy graniastos\u0142upa s\u0105 do siebie r\u00f3wnoleg\u0142e. Je\u015bli w podstawie b\u0119dziemy mieli wielok\u0105t foremny, wtedy graniastos\u0142up nazywamy prawid\u0142owym. Aby zrobi\u0107 czytelny rysunek graniastos\u0142upa postawy rysujemy \u201cuko\u015bnie\u201d.<\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">Poni\u017cej przyk\u0142ady graniastos\u0142up\u00f3w:<\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\"><img fetchpriority=\"high\" decoding=\"async\" class=\"wp-image-983 aligncenter\" src=\"http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/graniastos%C5%82upy-300x91.png\" sizes=\"(max-width: 501px) 100vw, 501px\" srcset=\"http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/graniastos\u0142upy-300x91.png 300w, http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/graniastos\u0142upy-768x232.png 768w, http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/graniastos\u0142upy.png 835w\" alt=\"\" width=\"501\" height=\"152\" \/><\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\"><span style=\"color: #4472c4;\"><strong>Graniastos\u0142up prosty<\/strong><\/span>\u00a0ma dwie takie same podstawy, kt\u00f3re s\u0105 r\u00f3wnoleg\u0142e. W jego podstawie mo\u017ce by\u0107 dowolny wielok\u0105t. \u015aciany boczne graniastos\u0142upa prostego s\u0105 prostok\u0105tami i s\u0105 one prostopad\u0142e do podstaw.<\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" class=\"alignnone wp-image-1286 aligncenter\" src=\"http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/siatka1-228x300.png\" sizes=\"(max-width: 180px) 100vw, 180px\" srcset=\"http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/siatka1-228x300.png 228w, http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/siatka1.png 279w\" alt=\"\" width=\"180\" height=\"237\" \/><\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">Je\u015bli w podstawie graniastos\u0142upa prostego b\u0119dzie prostok\u0105t, to taki graniastos\u0142up nazywamy\u00a0<strong>prostopad\u0142o\u015bcianem<\/strong>. W bry\u0142ach tych ka\u017cd\u0105 \u015bcian\u0119 mo\u017cna uzna\u0107 za podstaw\u0119.<\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" class=\"wp-image-1289 aligncenter\" src=\"http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/prostopad%C5%82o%C5%9Bcian-256x300.png\" sizes=\"(max-width: 172px) 100vw, 172px\" srcset=\"http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/prostopad\u0142o\u015bcian-256x300.png 256w, http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/prostopad\u0142o\u015bcian.png 374w\" alt=\"\" width=\"172\" height=\"202\" \/><\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">Je\u015bli wszystkie \u015bciany prostopad\u0142o\u015bcianu b\u0119d\u0105 kwadratami to taki graniastos\u0142up nazywamy\u00a0<strong>sze\u015bcianem<\/strong>.<\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1290 aligncenter\" src=\"http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/sze%C5%9Bcianek-300x282.png\" sizes=\"(max-width: 199px) 100vw, 199px\" srcset=\"http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/sze\u015bcianek-300x282.png 300w, http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/sze\u015bcianek.png 318w\" alt=\"\" width=\"199\" height=\"186\" \/><\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">W ka\u017cdym graniastos\u0142upie prostym mamy:<\/span><\/p>\n<ul>\n<li><span style=\"font-family: 'book antiqua', palatino, serif;\">dwie postawy b\u0119d\u0105ce wielok\u0105tami;<\/span><\/li>\n<li><span style=\"font-family: 'book antiqua', palatino, serif;\">\u015bciany boczne prostopad\u0142e do podstaw;<\/span><\/li>\n<li><span style=\"font-family: 'book antiqua', palatino, serif;\">odcinek, kt\u00f3ry \u0142\u0105czy podstawy i jest do nich prostopad\u0142y nazywany\u00a0<strong>wysoko\u015bci\u0105 graniastos\u0142upa.<\/strong><\/span><\/li>\n<\/ul>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\"><img loading=\"lazy\" decoding=\"async\" class=\" wp-image-1288 aligncenter\" src=\"http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/graniastos%C5%82up-opisy-300x189.png\" sizes=\"(max-width: 425px) 100vw, 425px\" srcset=\"http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/graniastos\u0142up-opisy-300x189.png 300w, http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/graniastos\u0142up-opisy.png 646w\" alt=\"\" width=\"425\" height=\"268\" \/><\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">Opr\u00f3cz graniastos\u0142up\u00f3w prostych mamy r\u00f3wnie\u017c <\/span><strong style=\"color: #4472c4; font-family: 'book antiqua', palatino, serif;\">graniastos\u0142upy pochy\u0142e<\/strong><span style=\"font-family: 'book antiqua', palatino, serif;\">, kt\u00f3rych \u015bciany boczne s\u0105 r\u00f3wnoleg\u0142obokami. Wysoko\u015b\u0107 nie jest kraw\u0119dzi\u0105 boczn\u0105, jak w graniastos\u0142upach prostych.<\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1466 aligncenter\" src=\"http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/GRANIASTOS%C5%81UP-pochy%C5%82y-300x116.png\" sizes=\"(max-width: 442px) 100vw, 442px\" srcset=\"http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/GRANIASTOS\u0141UP-pochy\u0142y-300x116.png 300w, http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/GRANIASTOS\u0141UP-pochy\u0142y.png 621w\" alt=\"\" width=\"442\" height=\"171\" \/><\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\"><span style=\"color: #4472c4;\"><strong>Przek\u0105tna graniastos\u0142upa<\/strong><\/span> jest to odcinek, kt\u00f3ry \u0142\u0105czy ze sob\u0105 dwa wierzcho\u0142ki graniastos\u0142upa i nie zawiera si\u0119 on w \u017cadnej ze \u015bcian.<\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1585 aligncenter\" src=\"http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/PRZEK\u0104TNE-GRANIASTOS\u0141UP\u00d3W-300x144.png\" alt=\"\" width=\"315\" height=\"151\" srcset=\"http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/PRZEK\u0104TNE-GRANIASTOS\u0141UP\u00d3W-300x144.png 300w, http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/PRZEK\u0104TNE-GRANIASTOS\u0141UP\u00d3W.png 681w\" sizes=\"(max-width: 315px) 100vw, 315px\" \/><\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">GRANIASTOS\u0141UP TR\u00d3JK\u0104TNY NIE MA \u017bADNEJ PRZEK\u0104TNEJ!<\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">W graniastos\u0142upie mamy r\u00f3wnie\u017c przek\u0105tn\u0105 podstawy czy przek\u0105tn\u0105 \u015bciany bocznej. Najwa\u017cniejsze jest aby nie myli\u0107 ze sob\u0105 przek\u0105tnych graniastos\u0142upa<\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1586 aligncenter\" src=\"http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/rodzaje-przek\u0105tnych-300x146.png\" alt=\"\" width=\"438\" height=\"213\" srcset=\"http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/rodzaje-przek\u0105tnych-300x146.png 300w, http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/rodzaje-przek\u0105tnych-768x374.png 768w, http:\/\/matma.com.pl\/wp-content\/uploads\/2020\/08\/rodzaje-przek\u0105tnych.png 826w\" sizes=\"(max-width: 438px) 100vw, 438px\" \/><\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\">Aby obliczy\u0107\u00a0<span style=\"color: #4472c4;\"><strong>pole powierzchni ca\u0142kowitej graniastos\u0142upa<\/strong><\/span>\u00a0musimy doda\u0107 do siebie pola wszystkich \u015bcian graniastos\u0142upa. Czyli jest to suma p\u00f3l dw\u00f3ch podstaw oraz wszystkich \u015bcian bocznych.<\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" class=\"aligncenter\" src=\"http:\/\/latex.codecogs.com\/gif.latex?P_{C}=2\\cdot&amp;space;P_{P}+P_{B}\" alt=\"P_{C}=2\\cdot P_{P}+P_{B}\" align=\"absmiddle\" \/><\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?P_{P}\" alt=\"P_{P}\" align=\"absmiddle\" \/>\u00a0to pole powierzchni podstawy.<\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?P_{B}\" alt=\"P_{B}\" align=\"absmiddle\" \/>\u00a0to pole powierzchni bocznej czyli suma p\u00f3l wszystkich \u015bcian bocznych.<\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\"><strong><span style=\"color: #4472c4;\">Obj\u0119to\u015b\u0107 graniastos\u0142upa<\/span>,\u00a0<\/strong>to iloczyn jego pola podstawy i d\u0142ugo\u015bci wysoko\u015bci graniastos\u0142upa<\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" class=\"aligncenter\" src=\"http:\/\/latex.codecogs.com\/gif.latex?V=P_{P}\\cdot&amp;space;H\" alt=\"V=P_{P}\\cdot H\" align=\"absmiddle\" \/><\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?P_{P}\" alt=\"P_{P}\" align=\"absmiddle\" \/>\u00a0to pole powierzchni podstawy.<\/span><\/p>\n<p><span style=\"font-family: 'book antiqua', palatino, serif;\"><img decoding=\"async\" src=\"http:\/\/latex.codecogs.com\/gif.latex?H\" alt=\"H\" align=\"absmiddle\" \/>\u00a0to wysoko\u015b\u0107 graniastos\u0142upa.<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Graniastos\u0142up\u00a0to bry\u0142a przestrzenna, kt\u00f3rego podstaw\u0105 jest wielok\u0105t, a \u015bciany boczne s\u0105 r\u00f3wnoleg\u0142obokami. Obie podstawy graniastos\u0142upa s\u0105 do siebie r\u00f3wnoleg\u0142e. Je\u015bli w podstawie b\u0119dziemy mieli wielok\u0105t foremny, wtedy graniastos\u0142up nazywamy prawid\u0142owym. Aby zrobi\u0107 czytelny rysunek graniastos\u0142upa postawy rysujemy \u201cuko\u015bnie\u201d. Poni\u017cej przyk\u0142ady graniastos\u0142up\u00f3w: Graniastos\u0142up prosty\u00a0ma dwie takie same podstawy, kt\u00f3re s\u0105 r\u00f3wnoleg\u0142e. W jego podstawie mo\u017ce by\u0107 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6,15],"tags":[],"class_list":["post-1584","post","type-post","status-publish","format-standard","hentry","category-baza-wiedzy","category-klasa-viii"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v20.1 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Graniastos\u0142upy. - Matma.com.pl<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/matma.com.pl\/?p=1584\" \/>\n<meta property=\"og:locale\" content=\"pl_PL\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Graniastos\u0142upy. - Matma.com.pl\" \/>\n<meta property=\"og:description\" content=\"Graniastos\u0142up\u00a0to bry\u0142a przestrzenna, kt\u00f3rego podstaw\u0105 jest wielok\u0105t, a \u015bciany boczne s\u0105 r\u00f3wnoleg\u0142obokami. Obie podstawy graniastos\u0142upa s\u0105 do siebie r\u00f3wnoleg\u0142e. Je\u015bli w podstawie b\u0119dziemy mieli wielok\u0105t foremny, wtedy graniastos\u0142up nazywamy prawid\u0142owym. Aby zrobi\u0107 czytelny rysunek graniastos\u0142upa postawy rysujemy \u201cuko\u015bnie\u201d. Poni\u017cej przyk\u0142ady graniastos\u0142up\u00f3w: Graniastos\u0142up prosty\u00a0ma dwie takie same podstawy, kt\u00f3re s\u0105 r\u00f3wnoleg\u0142e. 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