{"id":2945,"date":"2026-10-21T07:20:11","date_gmt":"2026-10-21T06:20:11","guid":{"rendered":"https:\/\/wpcalc.com\/pl\/matematyka\/kalkulator-trojkat-prostokatny\/"},"modified":"2026-10-21T07:20:11","modified_gmt":"2026-10-21T06:20:11","slug":"kalkulator-trojkat-prostokatny","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/pl\/matematyka\/kalkulator-trojkat-prostokatny\/","title":{"rendered":"Kalkulator tr\u00f3jk\u0105ta prostok\u0105tnego"},"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\">Rozwi\u0105\u017c tr\u00f3jk\u0105t prostok\u0105tny na podstawie dw\u00f3ch pomiar\u00f3w<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"2945\" data-favorite-title=\"Tr\u00f3jk\u0105t prostok\u0105tny\" data-favorite-url=\"https:\/\/wpcalc.com\/pl\/matematyka\/kalkulator-trojkat-prostokatny\/\" data-favorite-excerpt=\"Rozwi\u0105\u017c tr\u00f3jk\u0105t prostok\u0105tny, znaj\u0105c dwa elementy, w tym co najmniej jeden bok. Wpisz d\u0142ugo\u015bci bok\u00f3w, k\u0105t ostry, pole lub obw\u00f3d, aby obliczy\u0107 pozosta\u0142e warto\u015bci.\" 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-trojkat-prostokatny\/\" method=\"POST\" class=\"calculator\" id=\"calculator-2945\" data-post=\"2945\"><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\">Przyprostok\u0105tna 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><input type=\"text\" name=\"przyprostoktna_a\" id=\"field-1\" aria-describedby=\"calc-info-1\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" min=\"0\" value=\"3\"\/><small id=\"calc-info-1\">K\u0105t le\u017c\u0105cy naprzeciw. Podaj d\u0142ugo\u015bci w tej samej jednostce.<\/small><\/div><div class=\"field has-term\" id=\"input-2\"><label for=\"field-2\">Przyprostok\u0105tna b<\/label><div class=\"term\">b<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"przyprostoktna_b\" id=\"field-2\" aria-describedby=\"calc-info-2\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" min=\"0\" value=\"4\"\/><small id=\"calc-info-2\">K\u0105t le\u017c\u0105cy naprzeciw.<\/small><\/div><div class=\"field has-term\" id=\"input-3\"><label for=\"field-3\">Przeciwprostok\u0105tna c<\/label><div class=\"term\">c<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"przeciwprostoktna_c\" id=\"field-3\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" min=\"0\"\/><\/div><div class=\"field has-term\" id=\"input-4\"><label for=\"field-4\">K\u0105t \u03b1<\/label><div class=\"term\">\u03b1<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"kt_\" id=\"field-4\" aria-describedby=\"calc-unit-4 calc-info-4\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" min=\"0\" max=\"90\"\/><div class=\"unit\" id=\"calc-unit-4\">\u00b0<\/div><small id=\"calc-info-4\">K\u0105t naprzeciw przyprostok\u0105tnej a; musi mie\u015bci\u0107 si\u0119 mi\u0119dzy 0\u00b0 a 90\u00b0.<\/small><\/div><div class=\"field has-term\" id=\"input-5\"><label for=\"field-5\">K\u0105t \u03b2<\/label><div class=\"term\">\u03b2<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"kt_\" id=\"field-5\" aria-describedby=\"calc-unit-5 calc-info-5\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" min=\"0\" max=\"90\"\/><div class=\"unit\" id=\"calc-unit-5\">\u00b0<\/div><small id=\"calc-info-5\">K\u0105t naprzeciw przyprostok\u0105tnej b; musi mie\u015bci\u0107 si\u0119 mi\u0119dzy 0\u00b0 a 90\u00b0.<\/small><\/div><div class=\"field has-term\" id=\"input-6\"><label for=\"field-6\">Pole powierzchni<\/label><div class=\"term\">A<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"pole_powierzchni\" id=\"field-6\" aria-describedby=\"calc-info-6\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" min=\"0\"\/><small id=\"calc-info-6\">Opcjonalna znana warto\u015b\u0107 w jednostkach kwadratowych.<\/small><\/div><div class=\"field has-term\" id=\"input-7\"><label for=\"field-7\">Obw\u00f3d<\/label><div class=\"term\">P<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"obwd\" id=\"field-7\" aria-describedby=\"calc-info-7\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" min=\"0\"\/><small id=\"calc-info-7\">Opcjonalna znana warto\u015b\u0107, w tej samej jednostce d\u0142ugo\u015bci co boki.<\/small><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Przelicz ponownie\" id=\"calculate-button\" data-post=\"2945\">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> Rozwi\u0105zany tr\u00f3jk\u0105t<\/h3><div class=\"result-box\"><div class=\"field-result\" id=\"output-1\"><span class=\"field-title\"><span>Przyprostok\u0105tna a<\/span><span class=\"result-term\">a<\/span><\/span><span class=\"field-value\" id=\"result-1\" data-unit=\"jedn.\"><\/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>Przyprostok\u0105tna b<\/span><span class=\"result-term\">b<\/span><\/span><span class=\"field-value\" id=\"result-2\" data-unit=\"jedn.\"><\/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>Przeciwprostok\u0105tna c<\/span><span class=\"result-term\">c<\/span><\/span><span class=\"field-value\" id=\"result-3\" data-unit=\"jedn.\"><\/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>K\u0105t \u03b1<\/span><span class=\"result-term\">\u03b1<\/span><\/span><span class=\"field-value\" id=\"result-4\" data-unit=\"\u00b0\"><\/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>K\u0105t \u03b2<\/span><span class=\"result-term\">\u03b2<\/span><\/span><span class=\"field-value\" id=\"result-5\" data-unit=\"\u00b0\"><\/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 \u03b1 w radianach<\/span><span class=\"result-term\">\u03b1<\/span><\/span><span class=\"field-value\" id=\"result-6\" data-unit=\"rad\"><\/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-7\"><span class=\"field-title\"><span>K\u0105t \u03b2 w radianach<\/span><span class=\"result-term\">\u03b2<\/span><\/span><span class=\"field-value\" id=\"result-7\" data-unit=\"rad\"><\/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-8\"><span class=\"field-title\"><span>Pole powierzchni<\/span><span class=\"result-term\">A<\/span><\/span><span class=\"field-value\" id=\"result-8\" data-unit=\"jedn.\u00b2\"><\/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-9\"><span class=\"field-title\"><span>Obw\u00f3d<\/span><span class=\"result-term\">P<\/span><\/span><span class=\"field-value\" id=\"result-9\" data-unit=\"jedn.\"><\/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-10\"><span class=\"field-title\"><span>Wysoko\u015b\u0107 opuszczona na przeciwprostok\u0105tn\u0105<\/span><span class=\"result-term\">h<\/span><\/span><span class=\"field-value\" id=\"result-10\" data-unit=\"jedn.\"><\/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-11\"><span class=\"field-title\"><span>Promie\u0144 okr\u0119gu wpisanego<\/span><span class=\"result-term\">r<\/span><\/span><span class=\"field-value\" id=\"result-11\" data-unit=\"jedn.\"><\/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-12\"><span class=\"field-title\"><span>Promie\u0144 okr\u0119gu opisanego<\/span><span class=\"result-term\">R<\/span><\/span><span class=\"field-value\" id=\"result-12\" data-unit=\"jedn.\"><\/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-13\"><span class=\"field-title\"><span>Wykorzystane pomiary<\/span><\/span><span class=\"field-value\" id=\"result-13\"><\/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-14\"><span class=\"field-title\"><span>Tr\u00f3jk\u0105t w skali<\/span><\/span><span class=\"field-value\" id=\"result-14\"><\/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=\"2945\" 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>Wzory dla tr\u00f3jk\u0105ta prostok\u0105tnego<\/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\\[\\begin{aligned}c&amp;=\\sqrt{a^2+b^2}, &amp; A&amp;=\\frac{ab}{2}, &amp; P&amp;=a+b+c\\\\ \\alpha&amp;=\\arctan\\!\\left(\\frac{a}{b}\\right), &amp; \\beta&amp;=90^\\circ-\\alpha\\\\ h&amp;=\\frac{ab}{c}, &amp; r&amp;=\\frac{ab}{P}, &amp; R&amp;=\\frac{c}{2}\\end{aligned}\\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Gdzie:<\/strong><\/p>\n<ul>\n<li>\\(a\\) i \\(b\\) \u2014 przyprostok\u0105tne, le\u017c\u0105ce odpowiednio naprzeciw k\u0105t\u00f3w \\(\\alpha\\) i \\(\\beta\\)<\/li>\n<li>\\(c\\) \u2014 przeciwprostok\u0105tna, le\u017c\u0105ca naprzeciw k\u0105ta prostego<\/li>\n<li>\\(A\\) \u2014 pole; \\(P\\) \u2014 obw\u00f3d<\/li>\n<li>\\(h\\) \u2014 wysoko\u015b\u0107 opuszczona z wierzcho\u0142ka k\u0105ta prostego na przeciwprostok\u0105tn\u0105<\/li>\n<li>\\(r\\) \u2014 promie\u0144 okr\u0119gu wpisanego; \\(R\\) \u2014 promie\u0144 okr\u0119gu opisanego<\/li>\n<\/ul>\n<p>Boki i k\u0105ty ostre wyznaczaj\u0105 twierdzenie Pitagorasa i trygonometria. Gdy tr\u00f3jk\u0105t jest ju\u017c rozwi\u0105zany, jego boki okre\u015blaj\u0105 r\u00f3wnie\u017c pole, obw\u00f3d, wysoko\u015b\u0107 i promienie okr\u0119g\u00f3w.<\/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>Tr\u00f3jk\u0105t prostok\u0105tny \u2013 przyk\u0142ad obliczenia<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Przyk\u0142ad:<\/strong> warto\u015bciami domy\u015blnymi s\u0105 przyprostok\u0105tna a = 3 oraz przyprostok\u0105tna b = 4. Oblicz pozosta\u0142e warto\u015bci.<\/p>\n<p>\\[c=\\sqrt{3^2+4^2}=5,\\qquad \\alpha=\\arctan\\!\\left(\\frac{3}{4}\\right)\\approx36.8699^\\circ,\\qquad \\beta\\approx53.1301^\\circ\\]<\/p>\n<p>\\[A=\\frac{3\\cdot4}{2}=6,\\qquad P=3+4+5=12\\]<\/p>\n<p>\\[h=\\frac{3\\cdot4}{5}=2.4,\\qquad r=\\frac{3\\cdot4}{12}=1,\\qquad R=\\frac{5}{2}=2.5\\]<\/p>\n<p><strong>Wynik:<\/strong> boki maj\u0105 d\u0142ugo\u015bci 3, 4 i 5 jednostek; k\u0105ty ostre s\u0105 w przybli\u017ceniu r\u00f3wne 36,8699\u00b0 i 53,1301\u00b0. Pole wynosi 6 jednostek kwadratowych, a obw\u00f3d 12 jednostek.<\/p>\n<p>\n\r\n<\/section>\n<h2>Jak korzysta\u0107 z kalkulatora<\/h2>\n<ol>\n<li>Wpisz dwa znane elementy, w tym co najmniej jeden bok. Pola bok\u00f3w to <strong>Przyprostok\u0105tna a<\/strong>, <strong>Przyprostok\u0105tna b<\/strong> i <strong>Przeciwprostok\u0105tna c<\/strong>. Przyprostok\u0105tna a le\u017cy naprzeciw k\u0105ta \u03b1, a przyprostok\u0105tna b naprzeciw k\u0105ta \u03b2.<\/li>\n<li>W obliczeniach z bokiem i k\u0105tem wpisz bok oraz <strong>K\u0105t \u03b1<\/strong> lub <strong>K\u0105t \u03b2<\/strong> w stopniach. Mo\u017cesz te\u017c po\u0142\u0105czy\u0107 bok z <strong>Polem<\/strong> lub <strong>Obwodem<\/strong>. Wszystkie d\u0142ugo\u015bci podawaj w tej samej jednostce, a pole w odpowiedniej jednostce kwadratowej.<\/li>\n<li>Je\u015bli domy\u015blne warto\u015bci w polach <strong>Przyprostok\u0105tna a<\/strong> i <strong>Przyprostok\u0105tna b<\/strong> nie nale\u017c\u0105 do twoich danych, usu\u0144 je. Pola, kt\u00f3rych nie chcesz u\u017cywa\u0107, zostaw puste.<\/li>\n<li>Wyniki oblicze\u0144 znajdziesz w sekcji <strong>Rozwi\u0105zany tr\u00f3jk\u0105t<\/strong>: d\u0142ugo\u015bci bok\u00f3w, k\u0105ty w stopniach i radianach, pole, obw\u00f3d, wysoko\u015b\u0107 opuszczon\u0105 na przeciwprostok\u0105tn\u0105 oraz promienie okr\u0119g\u00f3w wpisanego i opisanego. Sprawd\u017a sekcj\u0119 <strong>Wykorzystane dane<\/strong>, aby zobaczy\u0107, kt\u00f3re wpisy wyznaczy\u0142y wynik.<\/li>\n<\/ol>\n<h2>Co m\u00f3wi o tr\u00f3jk\u0105cie rozwi\u0105zany wynik<\/h2>\n<p>Tr\u00f3jk\u0105t prostok\u0105tny ma jeden k\u0105t 90\u00b0. Jego przyprostok\u0105tne przecinaj\u0105 si\u0119 przy tym k\u0105cie, a przeciwprostok\u0105tna le\u017cy naprzeciw niego. K\u0105ty ostre \u03b1 i \u03b2 sumuj\u0105 si\u0119 do 90\u00b0. Dwa odpowiednie elementy, z kt\u00f3rych co najmniej jeden jest bokiem, wyznaczaj\u0105 zwykle zar\u00f3wno kszta\u0142t, jak i rozmiar tr\u00f3jk\u0105ta. Same dwa k\u0105ty okre\u015blaj\u0105 kszta\u0142t, ale nie pozwalaj\u0105 obliczy\u0107 d\u0142ugo\u015bci bok\u00f3w.<\/p>\n<p>Przeciwprostok\u0105tn\u0105 mo\u017cna wyznaczy\u0107 z obu przyprostok\u0105tnych za pomoc\u0105 r\u00f3wno\u015bci \\(a^2+b^2=c^2\\). Je\u015bli znana jest przyprostok\u0105tna i k\u0105t ostry, \u0142\u0105cz\u0105 je z pozosta\u0142ymi bokami stosunki trygonometryczne, na przyk\u0142ad \\(\\tan\\alpha=a\/b\\) oraz \\(\\sin\\alpha=a\/c\\). Je\u015bli chcesz obliczy\u0107 tylko te zale\u017cno\u015bci, skorzystaj z <a href=\"https:\/\/wpcalc.com\/pl\/matematyka\/kalkulator-twierdzenia-pitagorasa\/\">Kalkulatora twierdzenia Pitagorasa<\/a> albo <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/trigonometry-calculator\/\">Kalkulatora trygonometrii<\/a>.<\/p>\n<p>Pole jest r\u00f3wne po\u0142owie iloczynu przyprostok\u0105tnych, poniewa\u017c pe\u0142ni\u0105 one rol\u0119 prostopad\u0142ej podstawy i wysoko\u015bci. Pokazana w wyniku <strong>wysoko\u015b\u0107 opuszczona na przeciwprostok\u0105tn\u0105<\/strong> to inna wysoko\u015b\u0107: biegnie od wierzcho\u0142ka k\u0105ta prostego do przeciwprostok\u0105tnej. Promie\u0144 okr\u0119gu wpisanego to promie\u0144 okr\u0119gu stycznego do wszystkich trzech bok\u00f3w, a promie\u0144 okr\u0119gu opisanego to promie\u0144 okr\u0119gu przechodz\u0105cego przez wszystkie trzy wierzcho\u0142ki. W tr\u00f3jk\u0105cie prostok\u0105tnym promie\u0144 okr\u0119gu opisanego jest zawsze r\u00f3wny po\u0142owie przeciwprostok\u0105tnej. Je\u015bli potrzebujesz pola wyznaczonego z innych danych tr\u00f3jk\u0105ta, u\u017cyj <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/triangle-area-calculator\/\">Kalkulatora pola tr\u00f3jk\u0105ta<\/a>.<\/p>\n<h2>Zasady wpisywania danych i praktyczne ograniczenia<\/h2>\n<p>Wpisuj wy\u0142\u0105cznie dodatnie, sko\u0144czone warto\u015bci. Ka\u017cdy k\u0105t ostry musi by\u0107 wi\u0119kszy od 0\u00b0 i mniejszy od 90\u00b0, a przeciwprostok\u0105tna musi by\u0107 d\u0142u\u017csza ni\u017c ka\u017cda ze znanych przyprostok\u0105tnych. Bok po\u0142\u0105czony z polem lub obwodem musi ponadto opisywa\u0107 tr\u00f3jk\u0105t prostok\u0105tny, kt\u00f3ry mo\u017ce istnie\u0107; kalkulator zg\u0142osi b\u0142\u0105d, je\u015bli tak nie jest.<\/p>\n<p>Kalkulator u\u017cywa tylko dw\u00f3ch wpisanych warto\u015bci. Bierze pierwsze dwa dost\u0119pne boki w kolejno\u015bci: przyprostok\u0105tna a, przyprostok\u0105tna b, przeciwprostok\u0105tna c. Gdy wpisano tylko jeden bok, \u0142\u0105czy go z pierwsz\u0105 dost\u0119pn\u0105 warto\u015bci\u0105 w kolejno\u015bci: k\u0105t \u03b1, k\u0105t \u03b2, pole, obw\u00f3d. Pozosta\u0142e wpisy s\u0105 pomijane, a nie sprawdzane pod k\u0105tem sp\u00f3jno\u015bci, dlatego usu\u0144 zb\u0119dne dane i zajrzyj do sekcji <strong>Wykorzystane dane<\/strong>.<\/p>\n<p>Przeciwprostok\u0105tna po\u0142\u0105czona z polem lub obwodem mo\u017ce dopuszcza\u0107 dwa przypisania etykiet temu samemu kszta\u0142towi: zamiana przyprostok\u0105tnych a i b zamienia r\u00f3wnie\u017c k\u0105ty \u03b1 i \u03b2. Wynik i rysunek w skali pokazuj\u0105 wersj\u0119 z \\(a\\leq b\\); notatka pod <strong>Wykorzystane dane<\/strong> wskazuje, kiedy mo\u017cliwe jest r\u00f3wnie\u017c zamienione przypisanie. Rysunek jest przeskalowany tak, aby pokaza\u0107 proporcje, a nie rzeczywiste wymiary na ekranie.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Rozwi\u0105\u017c tr\u00f3jk\u0105t prostok\u0105tny, znaj\u0105c dwa elementy, w tym co najmniej jeden bok. Wpisz d\u0142ugo\u015bci bok\u00f3w, k\u0105t ostry, pole lub obw\u00f3d, aby obliczy\u0107 pozosta\u0142e warto\u015bci.<\/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-2945","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\/2945","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=2945"}],"version-history":[{"count":0,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/posts\/2945\/revisions"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/media?parent=2945"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/categories?post=2945"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/tags?post=2945"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}