{"id":2657,"date":"2026-10-15T01:37:36","date_gmt":"2026-10-15T00:37:36","guid":{"rendered":"https:\/\/wpcalc.com\/pl\/matematyka\/pole-wielokata-foremnego\/"},"modified":"2026-10-15T01:37:36","modified_gmt":"2026-10-15T00:37:36","slug":"pole-wielokata-foremnego","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/pl\/matematyka\/pole-wielokata-foremnego\/","title":{"rendered":"Kalkulator pola wielok\u0105ta foremnego"},"content":{"rendered":"<section class=\"not-prose calculator-box\" aria-labelledby=\"calculator-title\"><div class=\"flex items-baseline sm:items-center justify-between gap-2 sm:gap-3\"><div class=\"flex flex-col sm:flex-row sm:items-center gap-2\"><span class=\"icon icon-calculator text-primary-700! text-base! dark:text-primary-300!\" aria-hidden=\"true\"><\/span><h2 id=\"calculator-title\" class=\"text-lg font-display font-bold\">Oblicz pole wielok\u0105ta foremnego<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"2657\" data-favorite-title=\"Pole wielok\u0105ta foremnego\" data-favorite-url=\"https:\/\/wpcalc.com\/pl\/matematyka\/pole-wielokata-foremnego\/\" data-favorite-excerpt=\"Oblicz pole dowolnego wielok\u0105ta foremnego, znaj\u0105c d\u0142ugo\u015b\u0107 boku, apotem\u0119 lub promie\u0144 okr\u0119gu opisanego. Wpisz liczb\u0119 bok\u00f3w i jeden znany wymiar, a otrzymasz pole, obw\u00f3d i k\u0105ty.\" aria-label=\"Dodaj do ulubionych\" data-favorite-icon=\"icon icon-geometry\">\n    <span class=\"icon icon-shape-star-empty\" aria-hidden=\"true\"><\/span>\n  <\/button>\n  <div class=\"calcdash-favorite__tip\" aria-hidden=\"true\">\n   <span class=\"favorite-tooltip\" id=\"favorite-tooltip\"><\/span>\n  <\/div>\n  <span class=\"sr-only\" role=\"status\" id=\"favorite-status\"><\/span>\n<\/div><\/div><form action=\"https:\/\/wpcalc.com\/pl\/matematyka\/pole-wielokata-foremnego\/\" method=\"POST\" class=\"calculator\" id=\"calculator-2657\" data-post=\"2657\"><fieldset class=\"fieldset-input is-2\"><legend class=\"sr-only\">Pola danych<\/legend><div class=\"field has-term\" id=\"input-1\"><label for=\"field-1\">Liczba bok\u00f3w (n)<\/label><div class=\"term\">n<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"liczba_bokw_n\" id=\"field-1\" aria-describedby=\"calc-info-1\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"6\" min=\"3\" max=\"10000\" value=\"6\"\/><small id=\"calc-info-1\">Liczba ca\u0142kowita r\u00f3wnych bok\u00f3w, co najmniej 3.<\/small><\/div><div class=\"field\" id=\"input-2\"><label for=\"field-2\">Znana wielko\u015b\u0107<\/label><div class=\"grid grid-cols-1\"><select id=\"field-2\" name=\"znana_wielko\"><option value=\"side\" selected>D\u0142ugo\u015b\u0107 boku (s)<\/option><option value=\"apothem\">Apotema (a)<\/option><option value=\"circumradius\">Promie\u0144 okr\u0119gu opisanego (R)<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><div class=\"field has-term\" id=\"input-3\"><label for=\"field-3\">D\u0142ugo\u015b\u0107 boku (s)<\/label><div class=\"term\">s<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><div class=\"field-group\"><input type=\"text\" name=\"dugo_boku_s\" id=\"field-3\" aria-describedby=\"calc-info-3\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"10\" min=\"0\" value=\"10\"\/><div class=\"grid grid-cols-1\"><select id=\"field-4\" name=\"jednostka\" aria-label=\"Jednostka\"><option value=\"m\">m<\/option><option value=\"cm\" selected>cm<\/option><option value=\"mm\">mm<\/option><option value=\"ft\">ft<\/option><option value=\"in\">in<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><small id=\"calc-info-3\">D\u0142ugo\u015b\u0107 jednego boku wielok\u0105ta.<\/small><\/div><div class=\"field has-term\" id=\"input-4\" hidden><label for=\"field-5\">Apotema (a)<\/label><div class=\"term\">a<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><div class=\"field-group\"><input type=\"text\" name=\"apotema_a\" id=\"field-5\" aria-describedby=\"calc-info-5\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"8.66\" min=\"0\" value=\"8.66\"\/><div class=\"grid grid-cols-1\"><select id=\"field-6\" name=\"jednostka\" aria-label=\"Jednostka\"><option value=\"m\">m<\/option><option value=\"cm\" selected>cm<\/option><option value=\"mm\">mm<\/option><option value=\"ft\">ft<\/option><option value=\"in\">in<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><small id=\"calc-info-5\">Odleg\u0142o\u015b\u0107 od \u015brodka do \u015brodka boku (promie\u0144 okr\u0119gu wpisanego).<\/small><\/div><div class=\"field has-term\" id=\"input-5\" hidden><label for=\"field-7\">Promie\u0144 okr\u0119gu opisanego (R)<\/label><div class=\"term\">R<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><div class=\"field-group\"><input type=\"text\" name=\"promie_okrgu_opisanego_r\" id=\"field-7\" aria-describedby=\"calc-info-7\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"10\" min=\"0\" value=\"10\"\/><div class=\"grid grid-cols-1\"><select id=\"field-8\" name=\"jednostka\" aria-label=\"Jednostka\"><option value=\"m\">m<\/option><option value=\"cm\" selected>cm<\/option><option value=\"mm\">mm<\/option><option value=\"ft\">ft<\/option><option value=\"in\">in<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><small id=\"calc-info-7\">Odleg\u0142o\u015b\u0107 od \u015brodka do wierzcho\u0142ka (promie\u0144 okr\u0119gu opisanego).<\/small><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Przelicz ponownie\" id=\"calculate-button\" data-post=\"2657\">Oblicz<\/button><button type=\"reset\">Wyczy\u015b\u0107<\/button><\/div><div class=\"field is-checkbox hidden!\" id=\"field-auto-calc\"><input type=\"checkbox\" id=\"auto-calc\"><label for=\"auto-calc\">Obliczaj automatycznie<\/label><small>Po w\u0142\u0105czeniu wynik aktualizuje si\u0119 automatycznie po zmianie dowolnej warto\u015bci.<\/small><\/div><div class=\"fieldset-result is-hidden\" aria-labelledby=\"result-title\" aria-live=\"polite\" role=\"region\"> <h3 class=\"result-title bg-gradient-to-r from-primary-50 to-gray-50 dark:from-primary-900 dark:to-gray-800\" id=\"result-title\"><span class=\"icon icon-s-pulse\" aria-hidden=\"true\"><\/span> Tw\u00f3j wielok\u0105t<\/h3><div class=\"result-box\"><div class=\"field-result\" id=\"output-1\"><span class=\"field-title\"><span>Pole<\/span><span class=\"result-term\">A<\/span><\/span><span class=\"field-value\" id=\"result-1\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Kopiuj wynik\" aria-label=\"Kopiuj wynik\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-2\"><span class=\"field-title\"><span>Obw\u00f3d<\/span><span class=\"result-term\">P<\/span><\/span><span class=\"field-value\" id=\"result-2\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Kopiuj wynik\" aria-label=\"Kopiuj wynik\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-3\"><span class=\"field-title\"><span>D\u0142ugo\u015b\u0107 boku<\/span><span class=\"result-term\">s<\/span><\/span><span class=\"field-value\" id=\"result-3\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Kopiuj wynik\" aria-label=\"Kopiuj wynik\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-4\"><span class=\"field-title\"><span>Apotema<\/span><span class=\"result-term\">a<\/span><\/span><span class=\"field-value\" id=\"result-4\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Kopiuj wynik\" aria-label=\"Kopiuj wynik\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-5\"><span class=\"field-title\"><span>Promie\u0144 okr\u0119gu opisanego<\/span><span class=\"result-term\">R<\/span><\/span><span class=\"field-value\" id=\"result-5\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Kopiuj wynik\" aria-label=\"Kopiuj wynik\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-6\"><span class=\"field-title\"><span>K\u0105t wewn\u0119trzny<\/span><span class=\"result-term\">\u03b1<\/span><\/span><span class=\"field-value\" id=\"result-6\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Kopiuj wynik\" aria-label=\"Kopiuj wynik\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result is-column\" id=\"output-7\"><span class=\"field-title\"><span>Kroki oblicze\u0144<\/span><\/span><span class=\"field-value\" id=\"result-7\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Kopiuj wynik\" aria-label=\"Kopiuj wynik\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><\/div><div class=\"result-actions\"><button type=\"button\" class=\"copy-link\"><svg viewBox=\"0 0 20 20\" fill=\"currentColor\" aria-hidden=\"true\"><path d=\"M12.232 4.232a2.5 2.5 0 0 1 3.536 3.536l-1.225 1.224a.75.75 0 0 0 1.061 1.06l1.224-1.224a4 4 0 0 0-5.656-5.656l-3 3a4 4 0 0 0 .225 5.865.75.75 0 0 0 .977-1.138 2.5 2.5 0 0 1-.142-3.667l3-3Z\"\/><path d=\"M11.603 7.963a.75.75 0 0 0-.977 1.138 2.5 2.5 0 0 1 .142 3.667l-3 3a2.5 2.5 0 0 1-3.536-3.536l1.225-1.224a.75.75 0 0 0-1.061-1.06l-1.224 1.224a4 4 0 1 0 5.656 5.656l3-3a4 4 0 0 0-.225-5.865Z\"\/><\/svg><span class=\"copy-link__text\">Kopiuj link do tego obliczenia<\/span><\/button><span class=\"sr-only\" role=\"status\" id=\"copy-link-status\"><\/span><\/div><\/div><a href=\"#respond\" class=\"hidden transition-opacity duration-300 opacity-0 w-50 text-sm justify-center items-center gap-2 px-4 py-2 rounded bg-gray-200 text-gray-700 hover:bg-gray-300\" id=\"leave-comment\"><span class=\"icon icon-comments\" aria-hidden=\"true\"><\/span>Zostaw komentarz<\/a><\/form><\/section><section id=\"calc-reactions\" class=\"not-prose hidden my-12 bg-gradient-to-r from-primary-50 to-gray-50 border border-indigo-100 rounded-xl px-6 py-4 shadow-sm\" data-post=\"2657\" aria-live=\"polite\" aria-labelledby=\"calc-reactions-title\"><p id=\"calc-reactions-title\" class=\"text-sm text-gray-500 text-center mb-2\">Jak oceniasz ten wynik?<\/p><div role=\"group\" aria-labelledby=\"calc-reactions-title\" class=\"grid grid-cols-3 sm:flex sm:flex-row sm:justify-around gap-4 sm:items-center sm:flex-wrap\"><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"like\"><span class=\"reaction-count\" data-reaction-count=\"like\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude00 <\/span><span class=\"reaction-description font-medium\">Lubi\u0119 to <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"helpful\"><span class=\"reaction-count\" data-reaction-count=\"helpful\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\udca1 <\/span><span class=\"reaction-description font-medium\">Pomocne <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"confused\"><span class=\"reaction-count\" data-reaction-count=\"confused\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude15 <\/span><span class=\"reaction-description font-medium\">Niejasne <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"disappointed\"><span class=\"reaction-count\" data-reaction-count=\"disappointed\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude1e <\/span><span class=\"reaction-description font-medium\">Rozczarowanie <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"inaccurate\"><span class=\"reaction-count\" data-reaction-count=\"inaccurate\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\u274c <\/span><span class=\"reaction-description font-medium\">Niedok\u0142adne <\/span><\/button><\/div><div id=\"reaction-message\" class=\"hidden mt-8 rounded-md border border-primary-100 bg-white\/50 backdrop-blur-sm px-4 py-2 text-sm text-center shadow-sm\"><\/div><\/section><div class=\"calcdash-ad not-prose calcdash-ad--calc_result calcdash-ad--in-article\"><ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-1721569815777345\" data-ad-slot=\"6317458308\" data-ad-layout=\"in-article\" data-ad-format=\"fluid\"><\/ins><script>(adsbygoogle = window.adsbygoogle || []).push({});<\/script><\/div>\n<style id=\"wpcoderpro-inline-1\">.formula-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section class=\"formula-box\" aria-labelledby=\"formula-title\">\r\n\t<div id=\"formula-title\" class=\"not-prose font-display mb-4\"><span class=\"icon icon-formula\" aria-hidden=\"true\"><\/span><h2>Pole wielok\u0105ta foremnego<\/h2><\/div>\r\n\t<figure class=\"not-prose formula\">\r\n\t\t<figcaption class=\"formula-title\">Wz\u00f3r<\/figcaption>\r\n\t\t<div class=\"text-base text-gray-800\" id=\"formula\">\n\\[ A = \\frac{n\\,s^2}{4\\tan(\\pi\/n)} = \\frac{1}{2}\\,P\\,a = \\frac{1}{2}\\,n\\,R^2\\sin\\!\\left(\\frac{2\\pi}{n}\\right) \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Gdzie:<\/strong><\/p>\n<ul>\n<li>\\(n\\) \u2014 liczba bok\u00f3w (\\(n \\ge 3\\))<\/li>\n<li>\\(s\\) \u2014 d\u0142ugo\u015b\u0107 jednego boku<\/li>\n<li>\\(a\\) \u2014 apotema (odleg\u0142o\u015b\u0107 od \u015brodka do \u015brodka boku, czyli promie\u0144 okr\u0119gu wpisanego)<\/li>\n<li>\\(R\\) \u2014 promie\u0144 okr\u0119gu opisanego (odleg\u0142o\u015b\u0107 od \u015brodka do wierzcho\u0142ka)<\/li>\n<li>\\(P = n \\cdot s\\) \u2014 obw\u00f3d<\/li>\n<li>\\(\\pi \\approx 3.14159\\)<\/li>\n<\/ul>\n<p>Wszystkie trzy wyra\u017cenia daj\u0105 to samo pole \u2014 wybierz to, kt\u00f3re pasuje do wymiaru, jaki znasz. \u015arodkowa posta\u0107, \\(A = \\tfrac{1}{2}Pa\\), to klasyczny wynik uzyskany przez podzia\u0142 wielok\u0105ta na \\(n\\) identycznych tr\u00f3jk\u0105t\u00f3w, ka\u017cdy o podstawie \\(s\\) i wysoko\u015bci \\(a\\).<\/p>\n<p>\r\n<\/section>\n<style id=\"wpcoderpro-inline-2\">.calculation-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section id=\"calculation\" class=\"calculation-box\" aria-labelledby=\"calculation-title\">\r\n\t<div id=\"calculation-title\" class=\"not-prose font-display\"><span class=\"icon icon-unordered-list\" aria-hidden=\"true\"><\/span><h2>Pole wielok\u0105ta foremnego \u2013 przyk\u0142ad oblicze\u0144<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Przyk\u0142ad:<\/strong> sze\u015bciok\u0105t foremny o boku 10 cm.<\/p>\n<p>\\[ a = \\frac{s}{2\\tan(\\pi\/n)} = \\frac{10}{2\\tan(\\pi\/6)} = \\frac{10}{2 \\cdot 0.5774} \\approx 8.660 \\text{ cm} \\]<\/p>\n<p>\\[ P = n \\cdot s = 6 \\cdot 10 = 60 \\text{ cm} \\]<\/p>\n<p>\\[ A = \\tfrac{1}{2}\\,P\\,a = \\tfrac{1}{2} \\cdot 60 \\cdot 8.660 \\approx 259.81 \\text{ cm}^2 \\]<\/p>\n<p><strong>Wynik:<\/strong> pole wynosi oko\u0142o <strong>259,81 cm\u00b2<\/strong>. Kalkulator podaje tak\u017ce promie\u0144 okr\u0119gu opisanego \\(R = 10\\) cm oraz k\u0105t wewn\u0119trzny r\u00f3wny 120\u00b0.<\/p>\n<p>\n\r\n<\/section>\n<h2>Jak korzysta\u0107 z kalkulatora<\/h2>\n<ol>\n<li>Wpisz <strong>liczb\u0119 bok\u00f3w (n)<\/strong> \u2014 dowoln\u0105 liczb\u0119 ca\u0142kowit\u0105 od 3 (tr\u00f3jk\u0105t) do 10&nbsp;000.<\/li>\n<li>Na li\u015bcie rozwijanej <strong>Znana wielko\u015b\u0107<\/strong> wybierz wymiar, kt\u00f3ry znasz: <em>d\u0142ugo\u015b\u0107 boku (s)<\/em>, <em>apotem\u0119 (a)<\/em> lub <em>promie\u0144 okr\u0119gu opisanego (R)<\/em>. Pozostanie widoczne tylko pasuj\u0105ce pole.<\/li>\n<li>Wpisz warto\u015b\u0107 w polu, kt\u00f3re si\u0119 pojawi \u2014 <strong>d\u0142ugo\u015b\u0107 boku (s)<\/strong>, <strong>apotem\u0119 (a)<\/strong> lub <strong>promie\u0144 okr\u0119gu opisanego (R)<\/strong> \u2014 i wybierz jej jednostk\u0119 (m, cm, mm, ft lub in).<\/li>\n<li>Odczytaj wyniki: <strong>pole<\/strong> (w jednostce kwadratowej odpowiadaj\u0105cej wybranej jednostce), a tak\u017ce <strong>obw\u00f3d<\/strong>, <strong>d\u0142ugo\u015b\u0107 boku<\/strong>, <strong>apotem\u0119<\/strong>, <strong>promie\u0144 okr\u0119gu opisanego<\/strong>, <strong>k\u0105t wewn\u0119trzny<\/strong> i obliczenia krok po kroku.<\/li>\n<\/ol>\n<h2>Czym jest wielok\u0105t foremny?<\/h2>\n<p>Wielok\u0105t foremny to zamkni\u0119ta figura o prostoliniowych bokach, w kt\u00f3rej ka\u017cdy bok ma t\u0119 sam\u0105 d\u0142ugo\u015b\u0107, a wszystkie k\u0105ty wewn\u0119trzne s\u0105 r\u00f3wne. Znanymi przyk\u0142adami s\u0105 tr\u00f3jk\u0105t r\u00f3wnoboczny (\\(n=3\\)), kwadrat (\\(n=4\\)), pi\u0119ciok\u0105t foremny (\\(n=5\\)) i sze\u015bciok\u0105t foremny (\\(n=6\\)). Poniewa\u017c wszystkie boki i k\u0105ty s\u0105 r\u00f3wne, wielok\u0105t foremny jest w pe\u0142ni okre\u015blony zaledwie przez dwie liczby: liczb\u0119 bok\u00f3w i jeden wymiar liniowy.<\/p>\n<p>Wz\u00f3r na pole polega na poci\u0119ciu wielok\u0105ta na \\(n\\) przystaj\u0105cych tr\u00f3jk\u0105t\u00f3w spotykaj\u0105cych si\u0119 w \u015brodku figury. Ka\u017cdy tr\u00f3jk\u0105t ma podstaw\u0119 \\(s\\) (jeden bok wielok\u0105ta) i wysoko\u015b\u0107 \\(a\\) (apotem\u0119), wi\u0119c jego pole wynosi \\(\\tfrac{1}{2}sa\\). Po pomno\u017ceniu przez \\(n\\) bok\u00f3w otrzymujemy \\(A = \\tfrac{1}{2}na s = \\tfrac{1}{2}Pa\\). To ta sama logika, kt\u00f3r\u0105 stosuje si\u0119 dla pola ko\u0142a w granicy: gdy \\(n\\) staje si\u0119 bardzo du\u017ce, wielok\u0105t staje si\u0119 nieodr\u00f3\u017cnialny od ko\u0142a o promieniu \\(R\\), a wz\u00f3r d\u0105\u017cy do \\(\\pi R^2\\).<\/p>\n<p>Warto zrozumie\u0107 dwa pochodne wymiary. <strong>Apotema<\/strong> to promie\u0144 okr\u0119gu wpisanego \u2014 najwi\u0119kszego okr\u0119gu, jaki mie\u015bci si\u0119 w wielok\u0105cie, dotykaj\u0105cego ka\u017cdego boku w jego \u015brodku. <strong>Promie\u0144 okr\u0119gu opisanego<\/strong> to promie\u0144 okr\u0119gu przechodz\u0105cego przez wszystkie wierzcho\u0142ki. Te trzy wymiary wi\u0105\u017c\u0105 to\u017csamo\u015bci \\(s = 2a\\tan(\\pi\/n) = 2R\\sin(\\pi\/n)\\) oraz \\(a = R\\cos(\\pi\/n)\\) \u2014 dlatego kalkulator potrafi wyliczy\u0107 wszystkie z jednego dowolnego.<\/p>\n<p><strong>K\u0105t wewn\u0119trzny<\/strong> pokazywany w wynikach pochodzi ze wzoru \\(\\alpha = (n-2) \\cdot 180\u00b0\/n\\). Przydaje si\u0119 w budownictwie, przy uk\u0142adaniu p\u0142ytek i w stolarstwie: na przyk\u0142ad sze\u015bciok\u0105t ma naro\u017cniki 120\u00b0, wi\u0119c deski \u0142\u0105czysz, przycinaj\u0105c je pod k\u0105tem 60\u00b0.<\/p>\n<h2>Za\u0142o\u017cenia, ograniczenia i wskaz\u00f3wki<\/h2>\n<ul>\n<li><strong>Tylko wielok\u0105ty foremne.<\/strong> Wzory zak\u0142adaj\u0105, \u017ce wszystkie boki i k\u0105ty s\u0105 r\u00f3wne. Dla wielok\u0105ta nieforemnego podziel go na tr\u00f3jk\u0105ty lub trapezy i zsumuj pola.<\/li>\n<li><strong>Jednostki musz\u0105 by\u0107 sp\u00f3jne.<\/strong> Lista jednostek automatycznie przelicza wpisan\u0105 warto\u015b\u0107, a pole jest zwracane w odpowiadaj\u0105cej jednostce kwadratowej (np. cm \u2192 cm\u00b2). Je\u015bli liczysz r\u0119cznie, przed u\u017cyciem wzoru podaj wszystkie d\u0142ugo\u015bci w tej samej jednostce.<\/li>\n<li><strong>Wybierz wymiar, kt\u00f3ry faktycznie zmierzy\u0142e\u015b.<\/strong> W praktyce cz\u0119sto znasz odleg\u0142o\u015b\u0107 mi\u0119dzy przeciwleg\u0142ymi p\u0142aszczyznami (dwukrotno\u015b\u0107 apotemy przy parzystym \\(n\\)) albo promie\u0144 rozstawu otwor\u00f3w (promie\u0144 okr\u0119gu opisanego), a nie sam bok \u2014 wybierz t\u0119 opcj\u0119, \u017ceby unikn\u0105\u0107 b\u0142\u0119d\u00f3w zaokr\u0105gle\u0144 przy przeliczaniu po\u015brednim.<\/li>\n<li><strong>Du\u017ce \\(n\\).<\/strong> Przyjmowane s\u0105 warto\u015bci do 10&nbsp;000; dla takich figur wynik to w zasadzie pole ko\u0142a o promieniu \\(R\\), poniewa\u017c \\(A \\to \\pi R^2\\) przy \\(n \\to \\infty\\).<\/li>\n<li><strong>Zaokr\u0105glanie.<\/strong> Wyniki s\u0105 pokazywane do trzech miejsc po przecinku. Warto\u015bci po\u015brednie w krokach oblicze\u0144 s\u0105 zaokr\u0105glane do wy\u015bwietlenia, wi\u0119c r\u0119czne sprawdzenie mo\u017ce r\u00f3\u017cni\u0107 si\u0119 na ostatniej cyfrze.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Oblicz pole dowolnego wielok\u0105ta foremnego, znaj\u0105c d\u0142ugo\u015b\u0107 boku, apotem\u0119 lub promie\u0144 okr\u0119gu opisanego. Wpisz liczb\u0119 bok\u00f3w i jeden znany wymiar, a otrzymasz pole, obw\u00f3d i k\u0105ty.<\/p>\n","protected":false},"author":3168,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[6],"tags":[41],"class_list":["post-2657","post","type-post","status-publish","format-standard","hentry","category-matematyka","tag-geometria"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/posts\/2657","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/users\/3168"}],"replies":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/comments?post=2657"}],"version-history":[{"count":1,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/posts\/2657\/revisions"}],"predecessor-version":[{"id":2768,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/posts\/2657\/revisions\/2768"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/media?parent=2657"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/categories?post=2657"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/tags?post=2657"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}