{"id":2567,"date":"2026-10-14T19:35:01","date_gmt":"2026-10-14T18:35:01","guid":{"rendered":"https:\/\/wpcalc.com\/pl\/matematyka\/kalkulator-odleglosci-miedzy-punktami\/"},"modified":"2026-10-14T19:35:01","modified_gmt":"2026-10-14T18:35:01","slug":"kalkulator-odleglosci-miedzy-punktami","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/pl\/matematyka\/kalkulator-odleglosci-miedzy-punktami\/","title":{"rendered":"Kalkulator odleg\u0142o\u015bci mi\u0119dzy dwoma punktami"},"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 odleg\u0142o\u015b\u0107 mi\u0119dzy dwoma punktami<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"2567\" data-favorite-title=\"Odleg\u0142o\u015b\u0107 mi\u0119dzy dwoma punktami\" data-favorite-url=\"https:\/\/wpcalc.com\/pl\/matematyka\/kalkulator-odleglosci-miedzy-punktami\/\" data-favorite-excerpt=\"Wpisz wsp\u00f3\u0142rz\u0119dne dw\u00f3ch punkt\u00f3w, a obliczysz odleg\u0142o\u015b\u0107 w linii prostej mi\u0119dzy nimi, a tak\u017ce \u0394x, \u0394y i \u015brodek odcinka. Obs\u0142uguje metry, centymetry, stopy i cale.\" 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-odleglosci-miedzy-punktami\/\" method=\"POST\" class=\"calculator\" id=\"calculator-2567\" data-post=\"2567\"><fieldset class=\"fieldset-input is-2\"><legend class=\"sr-only\">Pola danych<\/legend><div class=\"is-single field\" id=\"input-1\"><span class=\"font-semibold\">Punkt 1<\/span><\/div><div class=\"field has-term\" id=\"input-2\"><label for=\"field-2\">x\u2081<\/label><div class=\"term\">x\u2081<\/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=\"x\" id=\"field-2\" aria-describedby=\"calc-info-2\" step=\"any\" value=\"1\"\/><div class=\"grid grid-cols-1\"><select id=\"field-3\" name=\"jednostka\" aria-label=\"Jednostka\"><option value=\"m\" selected>m<\/option><option value=\"cm\">cm<\/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-2\">Wsp\u00f3\u0142rz\u0119dna pozioma pierwszego punktu.<\/small><\/div><div class=\"field has-term\" id=\"input-3\"><label for=\"field-4\">y\u2081<\/label><div class=\"term\">y\u2081<\/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=\"y\" id=\"field-4\" aria-describedby=\"calc-info-4\" step=\"any\" value=\"2\"\/><div class=\"grid grid-cols-1\"><select id=\"field-5\" name=\"jednostka\" aria-label=\"Jednostka\"><option value=\"m\" selected>m<\/option><option value=\"cm\">cm<\/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-4\">Wsp\u00f3\u0142rz\u0119dna pionowa pierwszego punktu.<\/small><\/div><div class=\"is-single field\" id=\"input-4\"><span class=\"font-semibold\">Punkt 2<\/span><\/div><div class=\"field has-term\" id=\"input-5\"><label for=\"field-7\">x\u2082<\/label><div class=\"term\">x\u2082<\/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=\"x\" id=\"field-7\" aria-describedby=\"calc-info-7\" step=\"any\" value=\"4\"\/><div class=\"grid grid-cols-1\"><select id=\"field-8\" name=\"jednostka\" aria-label=\"Jednostka\"><option value=\"m\" selected>m<\/option><option value=\"cm\">cm<\/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\">Wsp\u00f3\u0142rz\u0119dna pozioma drugiego punktu.<\/small><\/div><div class=\"field has-term\" id=\"input-6\"><label for=\"field-9\">y\u2082<\/label><div class=\"term\">y\u2082<\/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=\"y\" id=\"field-9\" aria-describedby=\"calc-info-9\" step=\"any\" value=\"6\"\/><div class=\"grid grid-cols-1\"><select id=\"field-10\" name=\"jednostka\" aria-label=\"Jednostka\"><option value=\"m\" selected>m<\/option><option value=\"cm\">cm<\/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-9\">Wsp\u00f3\u0142rz\u0119dna pionowa drugiego punktu.<\/small><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Przelicz ponownie\" id=\"calculate-button\" data-post=\"2567\">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> Twoje wyniki<\/h3><div class=\"result-box\"><div class=\"field-result\" id=\"output-1\"><span class=\"field-title\"><span>Odleg\u0142o\u015b\u0107<\/span><span class=\"result-term\">d<\/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>R\u00f3\u017cnica pozioma (\u0394x)<\/span><span class=\"result-term\">\u0394x<\/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>R\u00f3\u017cnica pionowa (\u0394y)<\/span><span class=\"result-term\">\u0394y<\/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>\u015arodek odcinka<\/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><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=\"2567\" 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 odleg\u0142o\u015b\u0107 mi\u0119dzy dwoma punktami<\/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\\[ d = \\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Gdzie:<\/strong><\/p>\n<ul>\n<li>\\(d\\) \u2014 odleg\u0142o\u015b\u0107 mi\u0119dzy dwoma punktami<\/li>\n<li>\\(x_1, y_1\\) \u2014 wsp\u00f3\u0142rz\u0119dne pierwszego punktu<\/li>\n<li>\\(x_2, y_2\\) \u2014 wsp\u00f3\u0142rz\u0119dne drugiego punktu<\/li>\n<\/ul>\n<p>To wz\u00f3r na odleg\u0142o\u015b\u0107, b\u0119d\u0105cy bezpo\u015brednim zastosowaniem twierdzenia Pitagorasa: r\u00f3\u017cnica pozioma \\(\\Delta x = x_2 - x_1\\) i r\u00f3\u017cnica pionowa \\(\\Delta y = y_2 - y_1\\) tworz\u0105 przyprostok\u0105tne tr\u00f3jk\u0105ta prostok\u0105tnego, a odleg\u0142o\u015b\u0107 \\(d\\) jest jego przeciwprostok\u0105tn\u0105.<\/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>Odleg\u0142o\u015b\u0107 mi\u0119dzy dwoma punktami \u2014 przyk\u0142ad obliczenia<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Przyk\u0142ad:<\/strong> oblicz odleg\u0142o\u015b\u0107 mi\u0119dzy punktami \\(P_1 = (1, 2)\\) i \\(P_2 = (4, 6)\\), przy czym wszystkie wsp\u00f3\u0142rz\u0119dne podajemy w metrach (s\u0105 to domy\u015blne warto\u015bci w kalkulatorze).<\/p>\n<p>\\[ \\Delta x = 4 - 1 = 3, \\qquad \\Delta y = 6 - 2 = 4 \\]<\/p>\n<p>\\[ d = \\sqrt{3^2 + 4^2} = \\sqrt{9 + 16} = \\sqrt{25} = 5 \\]<\/p>\n<p><strong>Wynik:<\/strong> odleg\u0142o\u015b\u0107 wynosi \\(d = 5\\) m, przy czym \\(\\Delta x = 3\\) m, \\(\\Delta y = 4\\) m, a \u015brodek odcinka znajduje si\u0119 w punkcie \\((2.5,\\ 4.5)\\).<\/p>\n<p>\n\r\n<\/section>\n<h2>Jak korzysta\u0107 z kalkulatora<\/h2>\n<ol>\n<li>Wpisz <strong>x\u2081<\/strong> i <strong>y\u2081<\/strong> w polach <em>Punkt 1<\/em> \u2014 wsp\u00f3\u0142rz\u0119dn\u0105 poziom\u0105 i pionow\u0105 pierwszego punktu.<\/li>\n<li>Dla ka\u017cdej wsp\u00f3\u0142rz\u0119dnej wybierz jednostk\u0119 (<strong>m<\/strong>, <strong>cm<\/strong>, <strong>ft<\/strong> lub <strong>in<\/strong>) z listy rozwijanej obok pola wprowadzania.<\/li>\n<li>Wpisz <strong>x\u2082<\/strong> i <strong>y\u2082<\/strong> w polach <em>Punkt 2<\/em>, tak samo wybieraj\u0105c ich jednostki.<\/li>\n<li>Odczytaj wyniki: <strong>Odleg\u0142o\u015b\u0107<\/strong>, <strong>R\u00f3\u017cnica pozioma (\u0394x)<\/strong>, <strong>R\u00f3\u017cnica pionowa (\u0394y)<\/strong> oraz <strong>\u015arodek odcinka<\/strong>.<\/li>\n<\/ol>\n<p>Wynik jest zawsze podawany w jednostce wybranej dla x\u2081. Je\u015bli wpiszesz wsp\u00f3\u0142rz\u0119dne w r\u00f3\u017cnych jednostkach, kalkulator przeliczy wszystko wewn\u0119trznie na metry, a nast\u0119pnie zamieni wyniki z powrotem na jednostk\u0119 x\u2081 \u2014 mieszane dane wej\u015bciowe, takie jak \u201ex\u2081 w stopach, y\u2082 w calach\u201d, nadal zadzia\u0142aj\u0105 poprawnie.<\/p>\n<h2>Czym jest odleg\u0142o\u015b\u0107 mi\u0119dzy dwoma punktami?<\/h2>\n<p>Na p\u0142aszczy\u017anie dwuwymiarowej odleg\u0142o\u015b\u0107 mi\u0119dzy dwoma punktami to d\u0142ugo\u015b\u0107 prostego odcinka, kt\u00f3ry je \u0142\u0105czy \u2014 najkr\u00f3tsza mo\u017cliwa droga. Maj\u0105c punkty \\(P_1 = (x_1, y_1)\\) i \\(P_2 = (x_2, y_2)\\), mo\u017cesz wyobrazi\u0107 sobie tr\u00f3jk\u0105t prostok\u0105tny, kt\u00f3rego przyprostok\u0105tne to odcinek poziomy \\(\\Delta x\\) i pionowy \\(\\Delta y\\). Wz\u00f3r na odleg\u0142o\u015b\u0107 to po prostu twierdzenie Pitagorasa zastosowane do tego tr\u00f3jk\u0105ta \u2014 dlatego wystarczy podnie\u015b\u0107 do kwadratu, zsumowa\u0107 i wyci\u0105gn\u0105\u0107 pierwiastek.<\/p>\n<p>Wyniki po\u015brednie \u0394x i \u0394y bywaj\u0105 r\u00f3wnie przydatne jak sama odleg\u0142o\u015b\u0107. Pokazuj\u0105, jak daleko od siebie le\u017c\u0105 punkty wzd\u0142u\u017c ka\u017cdej osi i w kt\u00f3r\u0105 stron\u0119 \u2014 a to dok\u0142adnie to, czego potrzebujesz w zadaniach dotycz\u0105cych nachylenia, wektor\u00f3w czy przemieszczenia w fizyce. Nachylenie prostej przechodz\u0105cej przez oba punkty to na przyk\u0142ad \\(\\Delta y \/ \\Delta x\\).<\/p>\n<p>Kalkulator zwraca tak\u017ce <strong>\u015brodek<\/strong> odcinka: punkt \\(\\left(\\frac{x_1 + x_2}{2}, \\frac{y_1 + y_2}{2}\\right)\\), kt\u00f3ry le\u017cy dok\u0142adnie w po\u0142owie drogi mi\u0119dzy dwoma danymi punktami. Znajdziesz go, po prostu u\u015bredniaj\u0105c wsp\u00f3\u0142rz\u0119dne \u2014 ta sama idea stoi za <a href=\"https:\/\/wpcalc.com\/pl\/matematyka\/kalkulator-sredniej\/\">kalkulatorem \u015bredniej<\/a>.<\/p>\n<p>T\u0119 odleg\u0142o\u015b\u0107 nazywamy odleg\u0142o\u015bci\u0105 <em>euklidesow\u0105<\/em>, poniewa\u017c pochodzi ze zwyk\u0142ej geometrii euklidesowej. Jest to standardowa miara w geometrii analitycznej, nawigacji na p\u0142askich mapach, grafice komputerowej i wielu algorytmach uczenia maszynowego, kt\u00f3re por\u00f3wnuj\u0105 wektory cech.<\/p>\n<h2>Ograniczenia i praktyczne wskaz\u00f3wki<\/h2>\n<ul>\n<li><strong>Tylko 2D.<\/strong> Kalkulator dzia\u0142a na p\u0142aszczy\u017anie. Dla punkt\u00f3w tr\u00f3jwymiarowych \\((x, y, z)\\) trzeba by doda\u0107 trzeci sk\u0142adnik podniesiony do kwadratu: \\(d = \\sqrt{\\Delta x^2 + \\Delta y^2 + \\Delta z^2}\\).<\/li>\n<li><strong>Odleg\u0142o\u015b\u0107 w linii prostej.<\/strong> Wynik to bezpo\u015brednia odleg\u0142o\u015b\u0107 \u201ew linii powietrznej\u201d, a nie droga wzd\u0142u\u017c ulic czy z pomini\u0119ciem przeszk\u00f3d.<\/li>\n<li><strong>Ten sam uk\u0142ad wsp\u00f3\u0142rz\u0119dnych.<\/strong> Oba punkty musz\u0105 by\u0107 podane w tym samym uk\u0142adzie wsp\u00f3\u0142rz\u0119dnych (na przyk\u0142ad oba z tej samej siatki mapy). Mieszanie wsp\u00f3\u0142rz\u0119dnych z r\u00f3\u017cnych uk\u0142ad\u00f3w odniesienia daje wynik pozbawiony sensu.<\/li>\n<li><strong>Uwaga na jednostki.<\/strong> Wszystkie wyniki podawane s\u0105 w jednostce x\u2081. Je\u015bli chcesz otrzyma\u0107 odpowied\u017a w calach, przed obliczeniem ustaw jednostk\u0119 x\u2081 na <strong>in<\/strong>.<\/li>\n<li><strong>Wsp\u00f3\u0142rz\u0119dne ujemne nie s\u0105 problemem.<\/strong> R\u00f3\u017cnice \u0394x i \u0394y pokazywane s\u0105 jako warto\u015bci bezwzgl\u0119dne; sama odleg\u0142o\u015b\u0107 jest zawsze nieujemna.<\/li>\n<\/ul>\n<p>Szybka kontrola poprawno\u015bci: odleg\u0142o\u015b\u0107 nigdy nie mo\u017ce by\u0107 mniejsza ni\u017c \\(|\\Delta x|\\) ani \\(|\\Delta y|\\), a co najwy\u017cej r\u00f3wna si\u0119 \\(|\\Delta x| + |\\Delta y|\\). Je\u015bli wynik wykracza poza ten zakres, sprawd\u017a jeszcze raz wprowadzone wsp\u00f3\u0142rz\u0119dne.<\/p>\n<p>Je\u015bli podoba Ci si\u0119 konstrukcja tego obliczenia \u2014 pierwiastek z sumy kwadrat\u00f3w \u2014 polubisz te\u017c <a href=\"https:\/\/wpcalc.com\/pl\/matematyka\/kalkulator-sredniej-kwadratowej-rms\/\">kalkulator \u015bredniej kwadratowej (RMS)<\/a>, kt\u00f3ry stosuje ten sam schemat do listy liczb, oraz <a href=\"https:\/\/wpcalc.com\/pl\/matematyka\/pierwiastek-n-stopnia\/\">kalkulator pierwiastka n-tego stopnia<\/a> do innych pierwiastk\u00f3w ni\u017c pierwiastek kwadratowy.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Wpisz wsp\u00f3\u0142rz\u0119dne dw\u00f3ch punkt\u00f3w, a obliczysz odleg\u0142o\u015b\u0107 w linii prostej mi\u0119dzy nimi, a tak\u017ce \u0394x, \u0394y i \u015brodek odcinka. Obs\u0142uguje metry, centymetry, stopy i cale.<\/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-2567","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\/2567","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=2567"}],"version-history":[{"count":1,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/posts\/2567\/revisions"}],"predecessor-version":[{"id":2762,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/posts\/2567\/revisions\/2762"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/media?parent=2567"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/categories?post=2567"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/tags?post=2567"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}