{"id":2555,"date":"2026-10-14T18:10:42","date_gmt":"2026-10-14T17:10:42","guid":{"rendered":"https:\/\/wpcalc.com\/pl\/matematyka\/objetosc-i-pole-stozka\/"},"modified":"2026-10-14T18:10:42","modified_gmt":"2026-10-14T17:10:42","slug":"objetosc-i-pole-stozka","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/pl\/matematyka\/objetosc-i-pole-stozka\/","title":{"rendered":"Kalkulator obj\u0119to\u015bci i pola powierzchni sto\u017cka"},"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\">Obliczanie sto\u017cka na podstawie promienia oraz wysoko\u015bci lub tworz\u0105cej<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"2555\" data-favorite-title=\"Obj\u0119to\u015b\u0107 i pole powierzchni sto\u017cka\" data-favorite-url=\"https:\/\/wpcalc.com\/pl\/matematyka\/objetosc-i-pole-stozka\/\" data-favorite-excerpt=\"Oblicz obj\u0119to\u015b\u0107, tworz\u0105c\u0105, pole powierzchni bocznej, pole podstawy i pole powierzchni ca\u0142kowitej sto\u017cka prostego. Wpisz promie\u0144 oraz wysoko\u015b\u0107 albo d\u0142ugo\u015b\u0107 tworz\u0105cej.\" 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\/objetosc-i-pole-stozka\/\" method=\"POST\" class=\"calculator\" id=\"calculator-2555\" data-post=\"2555\"><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=\"3\" min=\"0\" value=\"3\"\/><div class=\"grid grid-cols-1\"><select id=\"field-2\" 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-1\">Promie\u0144 podstawy ko\u0142owej.<\/small><\/div><div class=\"field\" id=\"input-2\"><label for=\"field-3\">Drugi wymiar<\/label><div class=\"grid grid-cols-1\"><select id=\"field-3\" name=\"drugi_wymiar\"><option value=\"height\" selected>Wysoko\u015b\u0107 (h)<\/option><option value=\"slant\">Tworz\u0105ca (l)<\/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-4\">Wysoko\u015b\u0107 (h)<\/label><div class=\"term\">h<\/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=\"wysoko_h\" id=\"field-4\" aria-describedby=\"calc-info-4\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"4\" min=\"0\" value=\"4\"\/><div class=\"grid grid-cols-1\"><select id=\"field-5\" 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-4\">Odleg\u0142o\u015b\u0107 prostopad\u0142a od podstawy do wierzcho\u0142ka.<\/small><\/div><div class=\"field has-term\" id=\"input-4\" hidden><label for=\"field-6\">Tworz\u0105ca (l)<\/label><div class=\"term\">l<\/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=\"tworzca_l\" id=\"field-6\" aria-describedby=\"calc-info-6\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"5\" min=\"0\" value=\"5\"\/><div class=\"grid grid-cols-1\"><select id=\"field-7\" 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-6\">Odleg\u0142o\u015b\u0107 od kraw\u0119dzi podstawy do wierzcho\u0142ka wzd\u0142u\u017c powierzchni bocznej.<\/small><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Przelicz ponownie\" id=\"calculate-button\" data-post=\"2555\">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 sto\u017cek<\/h3><div class=\"result-box\"><div class=\"field-result\" id=\"output-1\"><span class=\"field-title\"><span>Obj\u0119to\u015b\u0107<\/span><span class=\"result-term\">V<\/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>Wysoko\u015b\u0107 (h)<\/span><span class=\"result-term\">h<\/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>Tworz\u0105ca<\/span><span class=\"result-term\">l<\/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>Pole powierzchni bocznej<\/span><span class=\"result-term\">A_L<\/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>Pole podstawy<\/span><span class=\"result-term\">A_B<\/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>Pole powierzchni ca\u0142kowitej<\/span><span class=\"result-term\">A<\/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=\"2555\" 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 obj\u0119to\u015b\u0107 i pole powierzchni sto\u017cka<\/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\\[ V = \\frac{1}{3}\\pi r^2 h \\qquad l = \\sqrt{r^2 + h^2} \\qquad h = \\sqrt{l^2 - r^2} \\\\\n A_L = \\pi r l \\qquad A_B = \\pi r^2 \\qquad A = \\pi r (r + l) \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Gdzie:<\/strong><\/p>\n<ul>\n<li>\\(r\\) \u2014 promie\u0144 ko\u0142owej podstawy<\/li>\n<li>\\(h\\) \u2014 wysoko\u015b\u0107 (prostopad\u0142a) od podstawy do wierzcho\u0142ka<\/li>\n<li>\\(l\\) \u2014 tworz\u0105ca sto\u017cka, czyli odleg\u0142o\u015b\u0107 mierzona wzd\u0142u\u017c powierzchni bocznej od kraw\u0119dzi podstawy do wierzcho\u0142ka<\/li>\n<li>\\(V\\) \u2014 obj\u0119to\u015b\u0107<\/li>\n<li>\\(A_L\\) \u2014 pole powierzchni bocznej<\/li>\n<li>\\(A_B\\) \u2014 pole podstawy<\/li>\n<li>\\(A\\) \u2014 pole powierzchni ca\u0142kowitej (bok + podstawa)<\/li>\n<\/ul>\n<p>Wzory te opisuj\u0105 sto\u017cek prosty o podstawie ko\u0142owej, kt\u00f3rego o\u015b jest prostopad\u0142a do podstawy. Twierdzenie Pitagorasa \u0142\u0105czy wysoko\u015b\u0107 i tworz\u0105c\u0105, wi\u0119c znajomo\u015b\u0107 jednej z tych wielko\u015bci wystarczy do wyznaczenia drugiej.<\/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>Obj\u0119to\u015b\u0107 i pole powierzchni sto\u017cka \u2014 przyk\u0142ad obliczenia<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Przyk\u0142ad:<\/strong> sto\u017cek o promieniu podstawy 3 cm i wysoko\u015bci 4 cm (domy\u015blne warto\u015bci w kalkulatorze).<\/p>\n<p>\\[ l = \\sqrt{r^2 + h^2} = \\sqrt{3^2 + 4^2} = \\sqrt{25} = 5 \\text{ cm} \\]<\/p>\n<p>\\[ V = \\frac{1}{3}\\pi r^2 h = \\frac{1}{3}\\pi \\cdot 3^2 \\cdot 4 = 12\\pi \\approx 37.699 \\text{ cm}^3 \\]<\/p>\n<p>\\[ A_L = \\pi r l = \\pi \\cdot 3 \\cdot 5 = 15\\pi \\approx 47.124 \\text{ cm}^2 \\]<\/p>\n<p>\\[ A_B = \\pi r^2 = \\pi \\cdot 3^2 = 9\\pi \\approx 28.274 \\text{ cm}^2 \\]<\/p>\n<p>\\[ A = A_L + A_B = 15\\pi + 9\\pi = 24\\pi \\approx 75.398 \\text{ cm}^2 \\]<\/p>\n<p><strong>Wynik:<\/strong> sto\u017cek ma obj\u0119to\u015b\u0107 oko\u0142o 37,699 cm\u00b3, tworz\u0105c\u0105 o d\u0142ugo\u015bci 5 cm, pole powierzchni bocznej 47,124 cm\u00b2, pole podstawy 28,274 cm\u00b2 oraz pole powierzchni ca\u0142kowitej 75,398 cm\u00b2.<\/p>\n<p>\n\r\n<\/section>\n<h2>Jak korzysta\u0107 z kalkulatora<\/h2>\n<ol>\n<li>Wpisz <strong>promie\u0144 (r)<\/strong> ko\u0142owej podstawy i wybierz jego jednostk\u0119 (m, cm, mm, ft lub in).<\/li>\n<li>Wybierz <strong>drugi wymiar<\/strong>: zaznacz <em>Wysoko\u015b\u0107 (h)<\/em>, je\u015bli znasz wysoko\u015b\u0107 prostopad\u0142\u0105, albo <em>Tworz\u0105c\u0105 (l)<\/em>, je\u015bli znasz odleg\u0142o\u015b\u0107 wzd\u0142u\u017c boku.<\/li>\n<li>Wpisz warto\u015b\u0107 w polu, kt\u00f3re si\u0119 pojawi \u2014 <strong>Wysoko\u015b\u0107 (h)<\/strong> lub <strong>Tworz\u0105ca (l)<\/strong> \u2014 i wybierz jej jednostk\u0119. Jednostki nie musz\u0105 by\u0107 takie same; kalkulator przelicza je automatycznie.<\/li>\n<li>Odczytaj wyniki: <strong>Obj\u0119to\u015b\u0107<\/strong>, <strong>Wysoko\u015b\u0107 (h)<\/strong>, <strong>Tworz\u0105ca<\/strong>, <strong>Pole powierzchni bocznej<\/strong>, <strong>Pole podstawy<\/strong> i <strong>Pole powierzchni ca\u0142kowitej<\/strong> \u2014 wszystkie wyra\u017cone w jednostce promienia \u2014 wraz z obliczeniami krok po kroku.<\/li>\n<\/ol>\n<h2>Czym jest sto\u017cek prosty?<\/h2>\n<p>Sto\u017cek prosty to bry\u0142a przestrzenna o ko\u0142owej podstawie i jednym wierzcho\u0142ku (szczycie), le\u017c\u0105cym dok\u0142adnie nad \u015brodkiem podstawy. Do jego opisu u\u017cywa si\u0119 dw\u00f3ch r\u00f3\u017cnych d\u0142ugo\u015bci, a ich mylenie to najcz\u0119stsze \u017ar\u00f3d\u0142o b\u0142\u0119d\u00f3w.<\/p>\n<p><strong>Wysoko\u015b\u0107<\/strong> \\(h\\) to prosta i prostopad\u0142a odleg\u0142o\u015b\u0107 od \u015brodka podstawy do wierzcho\u0142ka. <strong>Tworz\u0105ca<\/strong> \\(l\\) to odleg\u0142o\u015b\u0107 od dowolnego punktu na kraw\u0119dzi podstawy do wierzcho\u0142ka, mierzona wzd\u0142u\u017c uko\u015bnej powierzchni bocznej. Tworz\u0105ca jest zawsze d\u0142u\u017csza z tych dw\u00f3ch (albo r\u00f3wna wysoko\u015bci, gdy promie\u0144 wynosi zero), a obie wielko\u015bci \u0142\u0105czy twierdzenie Pitagorasa: \\( l^2 = r^2 + h^2 \\). Dlatego kalkulator potrzebuje tylko jednej z nich \u2014 drug\u0105 wyznacza automatycznie.<\/p>\n<p>Trzy warto\u015bci p\u00f3l s\u0105 podawane, bo odpowiadaj\u0105 na r\u00f3\u017cne praktyczne pytania. <strong>Pole podstawy<\/strong> pokazuje, ile materia\u0142u zajmuje ko\u0142owe dno. <strong>Pole powierzchni bocznej<\/strong> to powierzchnia zakrzywionego boku \u2014 na przyk\u0142ad ilo\u015b\u0107 materia\u0142u na \u015bciank\u0119 namiotu w kszta\u0142cie sto\u017cka albo etykieta owini\u0119ta wok\u00f3\u0142 sto\u017ckowego pojemnika. <strong>Pole powierzchni ca\u0142kowitej<\/strong> to suma obu warto\u015bci \u2014 przydaje si\u0119, gdy sto\u017cek jest w ca\u0142o\u015bci pokryty lub pomalowany ze wszystkich stron, jak ro\u017cek waflowy z p\u0142ask\u0105 warstw\u0105 lod\u00f3w na wierzchu.<\/p>\n<p>Obj\u0119to\u015b\u0107 interpretuje si\u0119 prosto: to ilo\u015b\u0107 miejsca, kt\u00f3r\u0105 sto\u017cek obejmuje. Warto zapami\u0119ta\u0107, \u017ce sto\u017cek mie\u015bci dok\u0142adnie jedn\u0105 trzeci\u0105 obj\u0119to\u015bci walca o tym samym promieniu podstawy i wysoko\u015bci \u2014 to w\u0142a\u015bnie czynnik \\( \\tfrac{1}{3} \\) we wzorze.<\/p>\n<h2>Za\u0142o\u017cenia, ograniczenia i wskaz\u00f3wki<\/h2>\n<ul>\n<li>Kalkulator zak\u0142ada <strong>sto\u017cek prosty<\/strong> o wierzcho\u0142ku le\u017c\u0105cym dok\u0142adnie nad \u015brodkiem podstawy. Sto\u017cki pochy\u0142e (z przechylonym wierzcho\u0142kiem) wymagaj\u0105 innych wzor\u00f3w na tworz\u0105c\u0105 i nie s\u0105 obs\u0142ugiwane.<\/li>\n<li>Je\u015bli podajesz tworz\u0105c\u0105, musi by\u0107 ona <strong>co najmniej tak d\u0142uga jak promie\u0144<\/strong> (\\( l \\ge r \\)). Tworz\u0105ca kr\u00f3tsza od promienia nie opisuje \u017cadnego rzeczywistego sto\u017cka i kalkulator wy\u015bwietli b\u0142\u0105d.<\/li>\n<li>Jednostki maj\u0105 znaczenie: obj\u0119to\u015b\u0107 podawana jest w jednostkach sze\u015bciennych (cm\u00b3, in\u00b3, \u2026), a pola w jednostkach kwadratowych (cm\u00b2, in\u00b2, \u2026), zawsze zgodnych z jednostk\u0105 wybran\u0105 dla promienia. Je\u015bli wysoko\u015b\u0107 jest podana w innej jednostce, jest najpierw przeliczana \u2014 wida\u0107 to w krokach oblicze\u0144.<\/li>\n<li>Wszystkie warto\u015bci wej\u015bciowe musz\u0105 by\u0107 dodatnie. Promie\u0144 lub wysoko\u015b\u0107 r\u00f3wna zeru sp\u0142aszczy\u0142aby sto\u017cek do ko\u0142a albo odcinka, daj\u0105c obj\u0119to\u015b\u0107 r\u00f3wn\u0105 zeru.<\/li>\n<li>Wyniki s\u0105 zaokr\u0105glane do trzech miejsc po przecinku; \\(\\pi\\) przyjmowane jest jako 3,14159, co jest wystarczaj\u0105co dok\u0142adne do praktycznie wszystkich zastosowa\u0144.<\/li>\n<\/ul>\n<p>Szybka kontrola przy obliczeniach r\u0119cznych: przy sta\u0142ym promieniu pole boczne \\( A_L = \\pi r l \\) ro\u015bnie wprost proporcjonalnie do tworz\u0105cej \u2014 to przyk\u0142ad <a href=\"https:\/\/wpcalc.com\/pl\/matematyka\/proporcjonalnosc-prosta\/\">proporcjonalno\u015bci prostej<\/a>. A pierwiastki w zale\u017cno\u015bciach mi\u0119dzy wysoko\u015bci\u0105 a tworz\u0105c\u0105 to po prostu szczeg\u00f3lny przypadek <a href=\"https:\/\/wpcalc.com\/pl\/matematyka\/pierwiastek-n-stopnia\/\">pierwiastka n-tego stopnia<\/a> przy \\(n = 2\\).<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Oblicz obj\u0119to\u015b\u0107, tworz\u0105c\u0105, pole powierzchni bocznej, pole podstawy i pole powierzchni ca\u0142kowitej sto\u017cka prostego. Wpisz promie\u0144 oraz wysoko\u015b\u0107 albo d\u0142ugo\u015b\u0107 tworz\u0105cej.<\/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-2555","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\/2555","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=2555"}],"version-history":[{"count":1,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/posts\/2555\/revisions"}],"predecessor-version":[{"id":2760,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/posts\/2555\/revisions\/2760"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/media?parent=2555"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/categories?post=2555"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/tags?post=2555"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}