{"id":2526,"date":"2026-10-14T07:24:11","date_gmt":"2026-10-14T06:24:11","guid":{"rendered":"https:\/\/wpcalc.com\/pl\/matematyka\/kalkulator-dlugosci-luku\/"},"modified":"2026-10-14T07:24:11","modified_gmt":"2026-10-14T06:24:11","slug":"kalkulator-dlugosci-luku","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/pl\/matematyka\/kalkulator-dlugosci-luku\/","title":{"rendered":"Kalkulator d\u0142ugo\u015bci \u0142uku"},"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\">D\u0142ugo\u015b\u0107 \u0142uku na podstawie promienia i k\u0105ta \u015brodkowego<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"2526\" data-favorite-title=\"D\u0142ugo\u015b\u0107 \u0142uku\" data-favorite-url=\"https:\/\/wpcalc.com\/pl\/matematyka\/kalkulator-dlugosci-luku\/\" data-favorite-excerpt=\"Wpisz promie\u0144 i k\u0105t \u015brodkowy okr\u0119gu (w stopniach, radianach lub gradach), a kalkulator natychmiast zwr\u00f3ci d\u0142ugo\u015b\u0107 \u0142uku wed\u0142ug wzoru  s  =  r \u03b8  wraz  z  pe\u0142nym  zapisem  oblicze\u0144.\" 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\/kalkulator-dlugosci-luku\/\" method=\"POST\" class=\"calculator\" id=\"calculator-2526\" data-post=\"2526\"><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\">Promie\u0144 (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_r\" id=\"field-1\" aria-describedby=\"calc-info-1\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"5\" min=\"0\" value=\"5\"\/><div class=\"grid grid-cols-1\"><select id=\"field-2\" name=\"jednostka\" aria-label=\"Jednostka\"><option value=\"m\" selected>m<\/option><option value=\"cm\">cm<\/option><option value=\"mm\">mm<\/option><option value=\"in\">in<\/option><option value=\"ft\">ft<\/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-1\">Odleg\u0142o\u015b\u0107 od \u015brodka okr\u0119gu do \u0142uku.<\/small><\/div><div class=\"field has-term\" id=\"input-2\"><label for=\"field-3\">K\u0105t \u015brodkowy (\u03b8)<\/label><div class=\"term\">\u03b8<\/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=\"number\" name=\"kt_rodkowy_\" id=\"field-3\" aria-describedby=\"calc-info-3\" placeholder=\"60\" step=\"any\" value=\"60\"\/><div class=\"grid grid-cols-1\"><select id=\"field-4\" name=\"jednostka\" aria-label=\"Jednostka\"><option value=\"deg\" selected>\u00b0<\/option><option value=\"rad\">rad<\/option><option value=\"grad\">grad<\/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\">K\u0105t mi\u0119dzy dwoma promieniami, kt\u00f3re wycinaj\u0105 \u0142uk.<\/small><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Przelicz ponownie\" id=\"calculate-button\" data-post=\"2526\">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> D\u0142ugo\u015b\u0107 Twojego \u0142uku<\/h3><div class=\"result-box\"><div class=\"field-result\" id=\"output-1\"><span class=\"field-title\"><span>D\u0142ugo\u015b\u0107 \u0142uku<\/span><span class=\"result-term\">s<\/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 is-column\" id=\"output-2\"><span class=\"field-title\"><span>Obliczenia<\/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><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=\"2526\" 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>Wz\u00f3r na d\u0142ugo\u015b\u0107 \u0142uku<\/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\\[ s = r \\cdot \\theta \\]\n\\[ \\theta_{\\text{rad}} = \\theta_{\\text{deg}} \\cdot \\frac{\\pi}{180} \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>gdzie:<\/strong><\/p>\n<ul>\n<li>\\(s\\) \u2014 d\u0142ugo\u015b\u0107 \u0142uku (w tej samej jednostce co promie\u0144)<\/li>\n<li>\\(r\\) \u2014 promie\u0144 okr\u0119gu<\/li>\n<li>\\(\\theta\\) \u2014 k\u0105t \u015brodkowy w <em>radianach<\/em> (k\u0105t mi\u0119dzy dwoma promieniami wyznaczaj\u0105cymi \u0142uk)<\/li>\n<\/ul>\n<p>Wz\u00f3r m\u00f3wi, \u017ce d\u0142ugo\u015b\u0107 \u0142uku jest wprost proporcjonalna zar\u00f3wno do promienia, jak i do k\u0105ta \u2014 podwojenie kt\u00f3rejkolwiek z tych warto\u015bci podwaja d\u0142ugo\u015b\u0107 \u0142uku. Je\u015bli k\u0105t masz podany w stopniach lub gradach, trzeba go najpierw przeliczy\u0107 na radiany, poniewa\u017c \\(1\\) radian definiuje si\u0119 jako k\u0105t, kt\u00f3remu odpowiada \u0142uk o d\u0142ugo\u015bci r\u00f3wnej promieniowi.<\/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>D\u0142ugo\u015b\u0107 \u0142uku \u2013 przyk\u0142ad obliczenia<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Przyk\u0142ad:<\/strong> promie\u0144 okr\u0119gu wynosi 5 m, a szukasz d\u0142ugo\u015bci \u0142uku wyznaczonego przez k\u0105t \u015brodkowy 60\u00b0.<\/p>\n<p>\\[ \\theta = 60 \\cdot \\frac{\\pi}{180} = \\frac{\\pi}{3} \\approx 1.0472 \\text{ rad} \\]<\/p>\n<p>\\[ s = r \\cdot \\theta = 5 \\times 1.0472 = 5.236 \\text{ m} \\]<\/p>\n<p><strong>Wynik:<\/strong> d\u0142ugo\u015b\u0107 \u0142uku wynosi oko\u0142o 5,236 m (dok\u0142adnie \\(\\frac{5\\pi}{3}\\) m, czyli oko\u0142o 17,5% ca\u0142ego obwodu \\(2\\pi r \\approx 31.416\\) m).<\/p>\n<p>\n\r\n<\/section>\n<h2>Jak korzysta\u0107 z kalkulatora d\u0142ugo\u015bci \u0142uku<\/h2>\n<ol>\n<li>Wpisz <strong>Promie\u0144 (r)<\/strong> \u2014 odleg\u0142o\u015b\u0107 od \u015brodka okr\u0119gu do \u0142uku \u2014 i wybierz jego jednostk\u0119 (m, cm, mm, cal lub stop\u0119).<\/li>\n<li>Wpisz <strong>K\u0105t \u015brodkowy (\u03b8)<\/strong> mi\u0119dzy dwoma promieniami wyznaczaj\u0105cymi \u0142uk i wybierz jego jednostk\u0119: <code>deg<\/code>, <code>rad<\/code> lub <code>grad<\/code>.<\/li>\n<li>Odczytaj <strong>D\u0142ugo\u015b\u0107 \u0142uku<\/strong> w panelu wynik\u00f3w \u2014 jest podana w tej samej jednostce co promie\u0144.<\/li>\n<li>Zajrzyj do wiersza <strong>Obliczenie<\/strong>, aby zobaczy\u0107 przeliczenie k\u0105ta na radiany i mno\u017cenie \\(s = r\\theta\\) wykonane krok po kroku.<\/li>\n<\/ol>\n<h2>Czym jest d\u0142ugo\u015b\u0107 \u0142uku?<\/h2>\n<p>\u0141uk to cz\u0119\u015b\u0107 okr\u0119gu mi\u0119dzy dwoma punktami. <em>D\u0142ugo\u015b\u0107 \u0142uku<\/em> to droga, kt\u00f3r\u0105 by\u015b przeby\u0142, id\u0105c po okr\u0119gu od jednego ko\u0144ca \u0142uku do drugiego \u2014 a nie odleg\u0142o\u015b\u0107 w linii prostej mi\u0119dzy tymi punktami (to by\u0142aby ci\u0119ciwa).<\/p>\n<p>O rozmiarze \u0142uku decyduj\u0105 tylko dwie wielko\u015bci: promie\u0144 okr\u0119gu i wyznaczaj\u0105cy go k\u0105t \u015brodkowy. K\u0105t \u015brodkowy to k\u0105t o wierzcho\u0142ku w \u015brodku okr\u0119gu, utworzony przez dwa promienie poprowadzone do ko\u0144c\u00f3w \u0142uku. Poniewa\u017c pe\u0142ny okr\u0105g odpowiada \\(2\\pi\\) radianom, \u0142uk obejmuj\u0105cy ca\u0142y okr\u0105g ma d\u0142ugo\u015b\u0107 \\(2\\pi r\\) \u2014 to znany wz\u00f3r na obw\u00f3d okr\u0119gu \u2014 a ka\u017cdy mniejszy k\u0105t odpowiada proporcjonalnej cz\u0119\u015bci tej warto\u015bci.<\/p>\n<p>W\u0142a\u015bnie dlatego we wzorze \\(s = r\\theta\\) k\u0105t musi by\u0107 wyra\u017cony w radianach. Radian to dok\u0142adnie taki k\u0105t, kt\u00f3rego \u0142uk ma d\u0142ugo\u015b\u0107 r\u00f3wn\u0105 promieniowi, wi\u0119c \\(\\theta\\) radian\u00f3w znaczy dos\u0142ownie \u201e\\(\\theta\\) d\u0142ugo\u015bci promienia po krzywej\u201d. W stopniach ta sama zale\u017cno\u015b\u0107 wymaga dodatkowego czynnika \\(\\pi\/180\\), kt\u00f3rym kalkulator zajmuje si\u0119 automatycznie: wpisz k\u0105t w dowolnej z trzech obs\u0142ugiwanych jednostek, a przeliczenie wykona si\u0119 samo.<\/p>\n<p>D\u0142ugo\u015b\u0107 \u0142uku pojawia si\u0119 wsz\u0119dzie tam, gdzie mierzy si\u0119 zakrzywione drogi: droga przebyta na okr\u0105g\u0142ym torze, d\u0142ugo\u015b\u0107 zgi\u0119tej rury lub wygi\u0119tej blachy, kabel nawini\u0119ty na szpul\u0119 czy droga ko\u0144ca wskaz\u00f3wki zegara w danym czasie. To r\u00f3wnie\u017c podstawa ruchu krzywoliniowego w fizyce i ca\u0142ek krzywoliniowych w rachunku r\u00f3\u017cniczkowym.<\/p>\n<h2>Najcz\u0119stsze k\u0105ty \u015brodkowe i ich \u0142uki<\/h2>\n<table>\n<thead>\n<tr>\n<th>K\u0105t \u015brodkowy<\/th>\n<th>U\u0142amek okr\u0119gu<\/th>\n<th>D\u0142ugo\u015b\u0107 \u0142uku<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>30\u00b0<\/td>\n<td>1\/12<\/td>\n<td>\\(\\frac{\\pi r}{6} \\approx 0.524\\,r\\)<\/td>\n<\/tr>\n<tr>\n<td>45\u00b0<\/td>\n<td>1\/8<\/td>\n<td>\\(\\frac{\\pi r}{4} \\approx 0.785\\,r\\)<\/td>\n<\/tr>\n<tr>\n<td>60\u00b0<\/td>\n<td>1\/6<\/td>\n<td>\\(\\frac{\\pi r}{3} \\approx 1.047\\,r\\)<\/td>\n<\/tr>\n<tr>\n<td>90\u00b0<\/td>\n<td>1\/4<\/td>\n<td>\\(\\frac{\\pi r}{2} \\approx 1.571\\,r\\)<\/td>\n<\/tr>\n<tr>\n<td>180\u00b0<\/td>\n<td>1\/2<\/td>\n<td>\\(\\pi r \\approx 3.142\\,r\\)<\/td>\n<\/tr>\n<tr>\n<td>360\u00b0<\/td>\n<td>1<\/td>\n<td>\\(2\\pi r \\approx 6.283\\,r\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Zauwa\u017c, \u017ce przy k\u0105tach do 90\u00b0 \u0142uk jest tylko nieznacznie d\u0142u\u017cszy od promienia \u2014 geometria radian\u00f3w sprawia, \u017ce te liczby pozostaj\u0105 bardzo intuicyjne.<\/p>\n<h2>Za\u0142o\u017cenia, ograniczenia i wskaz\u00f3wki<\/h2>\n<ul>\n<li><strong>Tylko okr\u0105g.<\/strong> Wz\u00f3r \\(s = r\\theta\\) dotyczy \u0142uk\u00f3w okr\u0119gu. Dla elips i dowolnych krzywych d\u0142ugo\u015b\u0107 \u0142uku trzeba wyznaczy\u0107 metodami rachunku r\u00f3\u017cniczkowego (przez ca\u0142kowanie), nie tym narz\u0119dziem.<\/li>\n<li><strong>K\u0105ty wi\u0119ksze ni\u017c pe\u0142ny obr\u00f3t.<\/strong> Je\u015bli wpiszesz wi\u0119cej ni\u017c 360\u00b0 (albo \\(2\\pi\\) rad), kalkulator nadal zwr\u00f3ci \\(r\\theta\\); \u0142uk po prostu okr\u0105\u017ca okr\u0105g wi\u0119cej ni\u017c raz \u2014 panel oblicze\u0144 zaznaczy ci t\u0119 sytuacj\u0119.<\/li>\n<li><strong>Ta sama jednostka na wej\u015bciu i wyj\u015bciu.<\/strong> Wynik zawsze jest podawany w jednostce, kt\u00f3r\u0105 wybra\u0142e\u015b dla promienia. Je\u015bli potrzebujesz odpowiedzi w innej jednostce, przelicz j\u0105 wcze\u015bniej.<\/li>\n<li><strong>Wycinek ko\u0142a a \u0142uk.<\/strong> To narz\u0119dzie mierzy tylko zakrzywion\u0105 kraw\u0119d\u017a. Je\u015bli potrzebujesz pola wycinka ko\u0142a ograniczonego \u0142ukiem i dwoma promieniami, u\u017cyj wzoru \\(A = \\tfrac{1}{2}r^2\\theta\\).<\/li>\n<li><strong>K\u0105ty ujemne.<\/strong> K\u0105t ujemny traktowany jest jako kierunek (zgodny z ruchem wskaz\u00f3wek zegara), a d\u0142ugo\u015b\u0107 \u0142uku wychodzi ujemna; je\u015bli interesuje ci\u0119 tylko warto\u015b\u0107 liczbow\u0105, wpisz warto\u015b\u0107 bezwzgl\u0119dn\u0105.<\/li>\n<\/ul>\n<h2>Poj\u0119cia pokrewne<\/h2>\n<p>D\u0142ugo\u015b\u0107 \u0142uku to klasyczny przyk\u0142ad <a href=\"https:\/\/wpcalc.com\/pl\/matematyka\/proporcjonalnosc-prosta\/\">proporcjonalno\u015bci prostej<\/a>: \\(s\\) jest wprost proporcjonalne zar\u00f3wno do \\(r\\), jak i do \\(\\theta\\), wi\u0119c zale\u017cno\u015b\u0107 daje prost\u0105 proporcj\u0119. Ko\u0144cowe mno\u017cenie \\(r \\times \\theta_{\\text{rad}}\\) wykonane r\u0119cznie mo\u017cesz sprawdzi\u0107 <a href=\"https:\/\/wpcalc.com\/pl\/matematyka\/kalkulator-mnozenia\/\">kalkulatorem mno\u017cenia<\/a>, a przeliczenie stopni na radiany to zwyk\u0142e <a href=\"https:\/\/wpcalc.com\/pl\/matematyka\/kalkulator-dzielenia\/\">dzielenie<\/a> przez \\(180\/\\pi\\).<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Wpisz promie\u0144 i k\u0105t \u015brodkowy okr\u0119gu (w stopniach, radianach lub gradach), a kalkulator natychmiast zwr\u00f3ci d\u0142ugo\u015b\u0107 \u0142uku wed\u0142ug wzoru  s  =  r \u03b8  wraz  z  pe\u0142nym  zapisem  oblicze\u0144.<\/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-2526","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\/2526","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=2526"}],"version-history":[{"count":1,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/posts\/2526\/revisions"}],"predecessor-version":[{"id":2714,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/posts\/2526\/revisions\/2714"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/media?parent=2526"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/categories?post=2526"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/tags?post=2526"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}