{"id":177463,"date":"2024-09-17T12:01:00","date_gmt":"2024-09-17T09:01:00","guid":{"rendered":"https:\/\/mathema.me\/?p=177463"},"modified":"2024-09-25T18:45:25","modified_gmt":"2024-09-25T15:45:25","slug":"wzor-herona-co-to-jest-i-jak-go-stosowac","status":"publish","type":"post","link":"https:\/\/mathema.me\/pl\/blog\/wzor-herona-co-to-jest-i-jak-go-stosowac\/","title":{"rendered":"Wz\u00f3r Herona: co to jest i jak go stosowa\u0107"},"content":{"rendered":"\n<p>Wz\u00f3r Herona s\u0142u\u017cy do obliczania pola tr\u00f3jk\u0105ta na podstawie d\u0142ugo\u015bci jego trzech bok\u00f3w \u2014 a, b, c. R\u00f3\u017cni si\u0119 od tradycyjnego sposobu, kt\u00f3ry wymaga znajomo\u015bci wysoko\u015bci, poniewa\u017c wykorzystuje jedynie d\u0142ugo\u015bci bok\u00f3w tr\u00f3jk\u0105ta.<\/p>\n\n\n\n<p>Wz\u00f3r wygl\u0105da tak:<\/p>\n\n\n\\[S=\\sqrt{p(p-a)(p-b)(p-c);}\\]\n\n\n\n<ul class=\"wp-block-list\">\n<li>S \u2014 pole tr\u00f3jk\u0105ta;<\/li>\n\n\n\n<li>a, b, c \u2014 d\u0142ugo\u015bci bok\u00f3w tr\u00f3jk\u0105ta;<\/li>\n\n\n\n<li>p \u2014 po\u0142owa obwodu tr\u00f3jk\u0105ta.<\/li>\n<\/ul>\n\n\n\n<p>Po\u0142owa obwodu to po\u0142owa sumy d\u0142ugo\u015bci wszystkich bok\u00f3w tr\u00f3jk\u0105ta. Jest oznaczana liter\u0105 p i obliczana wed\u0142ug wzoru:<\/p>\n\n\n\\[p=\\frac{a+b+c\u200b}2\\]\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Jak dzia\u0142a wz\u00f3r Herona?<\/strong><\/h2>\n\n\n\n<p>Wz\u00f3r Herona pozwala obliczy\u0107 pole dowolnego tr\u00f3jk\u0105ta, bez wzgl\u0119du na jego rodzaj (ostrok\u0105tny, prostok\u0105tny, rozwartok\u0105tny). Aby go zastosowa\u0107, wystarczy zna\u0107 d\u0142ugo\u015bci wszystkich bok\u00f3w tr\u00f3jk\u0105ta.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Przyk\u0142ad zastosowania wzoru Herona<\/strong><\/h2>\n\n\n\n<p>Za\u0142\u00f3\u017cmy, \u017ce mamy tr\u00f3jk\u0105t o bokach: a = 3, b = 4, c = 5. Naszym celem jest obliczenie jego pola.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"576\" src=\"https:\/\/i0.wp.com\/mathema.me\/wp-content\/uploads\/2024\/09\/formula-herona.png?resize=1024%2C576&#038;ssl=1\" alt=\"\" class=\"wp-image-175531\" srcset=\"https:\/\/i0.wp.com\/mathema.me\/wp-content\/uploads\/2024\/09\/formula-herona.png?resize=1024%2C576&amp;ssl=1 1024w, https:\/\/i0.wp.com\/mathema.me\/wp-content\/uploads\/2024\/09\/formula-herona.png?resize=300%2C169&amp;ssl=1 300w, https:\/\/i0.wp.com\/mathema.me\/wp-content\/uploads\/2024\/09\/formula-herona.png?resize=768%2C432&amp;ssl=1 768w, https:\/\/i0.wp.com\/mathema.me\/wp-content\/uploads\/2024\/09\/formula-herona.png?resize=50%2C28&amp;ssl=1 50w, https:\/\/i0.wp.com\/mathema.me\/wp-content\/uploads\/2024\/09\/formula-herona.png?resize=1600%2C900&amp;ssl=1 1600w, https:\/\/i0.wp.com\/mathema.me\/wp-content\/uploads\/2024\/09\/formula-herona.png?resize=1536%2C864&amp;ssl=1 1536w, https:\/\/i0.wp.com\/mathema.me\/wp-content\/uploads\/2024\/09\/formula-herona.png?resize=1200%2C675&amp;ssl=1 1200w, https:\/\/i0.wp.com\/mathema.me\/wp-content\/uploads\/2024\/09\/formula-herona.png?resize=600%2C338&amp;ssl=1 600w, https:\/\/i0.wp.com\/mathema.me\/wp-content\/uploads\/2024\/09\/formula-herona.png?w=1920&amp;ssl=1 1920w\" sizes=\"auto, (max-width: 1000px) 100vw, 1000px\" \/><\/figure>\n\n\n\n<p>Najpierw obliczamy po\u0142ow\u0119 obwodu tr\u00f3jk\u0105ta:<\/p>\n\n\n\\[p=\\frac{3+4+5\u200b}2=\\frac{12}2=6\\]\n\n\n\n<p>Nast\u0119pnie podstawiamy warto\u015bci <strong>p, a, b, c<\/strong> do wzoru Herona:<\/p>\n\n\n\\[S=\\sqrt{6(6-3)(6-4)(6-5)}=\\sqrt{6\\cdot3\\cdot2\\cdot1}=\\sqrt{36}=6\\]\n\n\n\n<p>Zatem pole tr\u00f3jk\u0105ta o bokach <strong>a = 3, b = 4, c = 5<\/strong> wynosi 6 cm\u00b2.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Kim by\u0142 Heron z Aleksandrii?<\/strong><\/h2>\n\n\n\n<p>Heron z Aleksandrii (ok. 10\u201370 n.e.) by\u0142 wybitnym staro\u017cytnym greckim matematykiem i in\u017cynierem, kt\u00f3ry przyczyni\u0142 si\u0119 do rozwoju matematyki, fizyki i in\u017cynierii. Dzia\u0142a\u0142 w Aleksandrii, jednym z najwa\u017cniejszych o\u015brodk\u00f3w naukowych staro\u017cytnego \u015bwiata.<\/p>\n\n\n\n<p>Heron zas\u0142yn\u0105\u0142 przede wszystkim odkryciem wzoru na obliczanie pola tr\u00f3jk\u0105ta, kt\u00f3ry dzi\u015b nosi jego imi\u0119. Opr\u00f3cz matematyki zajmowa\u0142 si\u0119 in\u017cynieri\u0105 i opisa\u0142 wiele urz\u0105dze\u0144 mechanicznych, takich jak prototypy maszyn parowych, automatyczne drzwi czy fontanny, kt\u00f3re ukazuj\u0105 jego niezwyk\u0142e umiej\u0119tno\u015bci techniczne.<\/p>\n\n\n\n<p>Heron mia\u0142 r\u00f3wnie\u017c du\u017cy wp\u0142yw na rozw\u00f3j mechaniki. W swoich dzie\u0142ach, takich jak &#8222;Mechanika&#8221; i &#8222;Pneumatica&#8221;, opisywa\u0142 urz\u0105dzenia, kt\u00f3re by\u0142y u\u017cywane w praktyce i inspirowa\u0142y wynalazc\u00f3w przez wieki.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Podsumowanie<\/strong><\/h2>\n\n\n\n<p>Wz\u00f3r Herona to uniwersalna metoda na obliczanie pola tr\u00f3jk\u0105ta, szczeg\u00f3lnie przydatna, gdy trudno jest znale\u017a\u0107 wysoko\u015b\u0107 tr\u00f3jk\u0105ta w inny spos\u00f3b. Pozwala on obliczy\u0107 pole ka\u017cdego tr\u00f3jk\u0105ta, znaj\u0105c jedynie d\u0142ugo\u015bci jego bok\u00f3w.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Wz\u00f3r Herona s\u0142u\u017cy do obliczania pola tr\u00f3jk\u0105ta na podstawie d\u0142ugo\u015bci jego trzech bok\u00f3w \u2014 a, b, c. R\u00f3\u017cni si\u0119 od [&hellip;]<\/p>\n","protected":false},"author":15,"featured_media":177428,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"_glsr_average":0,"_glsr_ranking":0,"_glsr_reviews":0,"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[37,3386],"tags":[],"class_list":["post-177463","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-blog","category-geometria"],"acf":[],"jetpack_featured_media_url":"https:\/\/i0.wp.com\/mathema.me\/wp-content\/uploads\/2024\/09\/post-cover-5-5.jpg?fit=1082%2C675&ssl=1","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/mathema.me\/pl\/wp-json\/wp\/v2\/posts\/177463","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mathema.me\/pl\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathema.me\/pl\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathema.me\/pl\/wp-json\/wp\/v2\/users\/15"}],"replies":[{"embeddable":true,"href":"https:\/\/mathema.me\/pl\/wp-json\/wp\/v2\/comments?post=177463"}],"version-history":[{"count":6,"href":"https:\/\/mathema.me\/pl\/wp-json\/wp\/v2\/posts\/177463\/revisions"}],"predecessor-version":[{"id":177470,"href":"https:\/\/mathema.me\/pl\/wp-json\/wp\/v2\/posts\/177463\/revisions\/177470"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/mathema.me\/pl\/wp-json\/wp\/v2\/media\/177428"}],"wp:attachment":[{"href":"https:\/\/mathema.me\/pl\/wp-json\/wp\/v2\/media?parent=177463"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathema.me\/pl\/wp-json\/wp\/v2\/categories?post=177463"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathema.me\/pl\/wp-json\/wp\/v2\/tags?post=177463"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}