{"id":18125,"date":"2020-09-28T16:17:42","date_gmt":"2020-09-28T19:17:42","guid":{"rendered":"https:\/\/beduka.com\/blog\/?p=18125"},"modified":"2023-04-03T16:59:38","modified_gmt":"2023-04-03T19:59:38","slug":"exercicios-de-geometria-espacial","status":"publish","type":"post","link":"https:\/\/beduka.com\/blog\/exercicios\/matematica-exercicios\/exercicios-de-geometria-espacial\/","title":{"rendered":"Exerc\u00edcios de Geometria Espacial com Gabarito"},"content":{"rendered":"\n<blockquote class=\"wp-block-quote\"><p><strong><em>Geometria espacial \u00e9 a \u00e1rea da Matem\u00e1tica respons\u00e1vel pela investiga\u00e7\u00e3o de s\u00f3lidos no espa\u00e7o, ou seja, ela analisa objetos tridimensionais. Essas formas s\u00e3o compostas por pontos, retas e planos. Leia nosso resumo e coloque em pr\u00e1tica todo o seu conhecimento com Exerc\u00edcios de Geometria Espacial. <\/em><\/strong><\/p><\/blockquote>\n\n\n\n<p>Quando voc\u00ea terminar os <strong>Exerc\u00edcios de Geometria Espacial<\/strong>, fa\u00e7a nosso <a href=\"https:\/\/beduka.com\/simulado-enem\/#\/\" target=\"_blank\" rel=\"noreferrer noopener\" aria-label=\" (abre numa nova aba)\">Simulado Enem<\/a>. Coloque em pr\u00e1tica todo o seu estudo, escolhendo as mat\u00e9rias que deseja treinar.&nbsp;<\/p>\n\n\n\n<p>Observe a diferen\u00e7a entre as duas imagens:&nbsp;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" loading=\"lazy\" width=\"600\" height=\"300\" src=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/09\/imagem-1-2.jpg\" alt=\"Qual a diferen\u00e7a entre 2D e 3D?\" class=\"wp-image-18137\" srcset=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/09\/imagem-1-2.jpg 600w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/09\/imagem-1-2-300x150.jpg 300w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/09\/imagem-1-2-320x160.jpg 320w\" sizes=\"(max-width: 600px) 100vw, 600px\" \/><\/figure><\/div>\n\n\n<ul><li>Perceba que a primeira casa n\u00e3o possui profundidade, \u00e9 o que chamamos de 2D (duas dimens\u00f5es);<\/li><\/ul>\n\n\n\n<ul><li>J\u00e1 a segunda casa possui tr\u00eas dimens\u00f5es, \u00e9 o que chamamos de 3D (3 dimens\u00f5es).<\/li><\/ul>\n\n\n\n<p><strong>Um objeto geom\u00e9trico espacial, \u00e9 como a casa em 3D, eles possuem profundidade.<\/strong> Logo, as suas medidas levam em conta as tr\u00eas dimens\u00f5es do objeto.<\/p>\n\n\n\n<p>Neste post n\u00f3s iremos estudar quais as caracter\u00edsticas de um objeto espacial e como analis\u00e1-los no vestibular.<\/p>\n\n\n\n<p>Para isso, ao final, selecionamos Exerc\u00edcios de Geometria Espacial dos melhores Vestibulares do Brasil.<\/p>\n\n\n\n<ul><li><a href=\"https:\/\/beduka.com\/blog\/materias\/matematica\/geometria-espacial\/\" target=\"_blank\" rel=\"noreferrer noopener\" aria-label=\" (abre numa nova aba)\">Entenda a Geometria Espacial e seus Principais Conceitos e F\u00f3rmulas!<\/a><\/li><\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">O que \u00e9 Geometria Espacial?<\/h2>\n\n\n\n<p><strong>Geometria espacial \u00e9 a \u00e1rea da Matem\u00e1tica respons\u00e1vel pela investiga\u00e7\u00e3o de s\u00f3lidos no espa\u00e7o<\/strong>, ou seja, ela analisa objetos tridimensionais.<strong> Essas formas s\u00e3o compostas por pontos, retas e planos.&nbsp;<\/strong><br><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" loading=\"lazy\" width=\"600\" height=\"300\" src=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/09\/imagem-2-4.jpg\" alt=\"O que s\u00e3o pontos, arestas e faces?\" class=\"wp-image-18136\" srcset=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/09\/imagem-2-4.jpg 600w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/09\/imagem-2-4-300x150.jpg 300w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/09\/imagem-2-4-320x160.jpg 320w\" sizes=\"(max-width: 600px) 100vw, 600px\" \/><\/figure><\/div>\n\n\n<h2 class=\"wp-block-heading\">Quais as Classifica\u00e7\u00f5es dos S\u00f3lidos Espaciais?<\/h2>\n\n\n\n<p>Os S\u00f3lidos Espaciais <strong>podem ser classificados em tr\u00eas tipos:<\/strong><br><\/p>\n\n\n\n<p><strong>Poliedros<\/strong>: s\u00e3o fechados e constitu\u00eddos de v\u00e9rtice, aresta e face.&nbsp;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" loading=\"lazy\" width=\"600\" height=\"300\" src=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/09\/prismas.jpg\" alt=\"Prismas\" class=\"wp-image-18134\" srcset=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/09\/prismas.jpg 600w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/09\/prismas-300x150.jpg 300w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/09\/prismas-320x160.jpg 320w\" sizes=\"(max-width: 600px) 100vw, 600px\" \/><\/figure><\/div>\n\n\n<p><strong>Corpos redondos<\/strong>: s\u00e3o aqueles que apresentam curvas e a sua medida espec\u00edfica \u00e9 o<strong> raio (r)<\/strong>.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" loading=\"lazy\" width=\"600\" height=\"300\" src=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/09\/cone-e-esfera.jpg\" alt=\"Cone e Esfera\" class=\"wp-image-18135\" srcset=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/09\/cone-e-esfera.jpg 600w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/09\/cone-e-esfera-300x150.jpg 300w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/09\/cone-e-esfera-320x160.jpg 320w\" sizes=\"(max-width: 600px) 100vw, 600px\" \/><\/figure><\/div>\n\n\n<p><strong>S\u00f3lidos de Plat\u00e3o<\/strong>: <strong>Os 5 s\u00f3lidos teorizados por Plat\u00e3o s\u00e3o<\/strong>: tetraedro (pir\u00e2mide), hexaedro (cubo), octaedro (8 faces), dodecaedro (12 faces) e icosaedro (20 lados).&nbsp;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" loading=\"lazy\" width=\"600\" height=\"300\" src=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/09\/s\u00f3lidos-de-plat\u00e3o.jpg\" alt=\"Os  poliedros de Plat\u00e3o\" class=\"wp-image-18133\" srcset=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/09\/s\u00f3lidos-de-plat\u00e3o.jpg 600w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/09\/s\u00f3lidos-de-plat\u00e3o-300x150.jpg 300w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/09\/s\u00f3lidos-de-plat\u00e3o-320x160.jpg 320w\" sizes=\"(max-width: 600px) 100vw, 600px\" \/><\/figure><\/div>\n\n\n<p>Esses poliedros s\u00e3o classificados assim, pois possuem arestas e faces congruentes.<strong> Tamb\u00e9m s\u00e3o chamados de poliedros regulares.<\/strong> Plat\u00e3o os analisou para tentar entender o surgimento do mundo atrav\u00e9s das formas geom\u00e9tricas. <\/p>\n\n\n\n<h2 class=\"wp-block-heading\">F\u00f3rmulas da Geometria Espacial<\/h2>\n\n\n\n<p>Na Geometria Espacial o objetivo do c\u00e1lculo \u00e9 descobrir o <strong>espa\u00e7o ocupado (volume)<\/strong> e os <strong>contornos (\u00e1rea total)<\/strong>, mas respeitando as dimens\u00f5es a mais que os s\u00f3lidos t\u00eam.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Cubo: volume e \u00e1rea total<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" loading=\"lazy\" width=\"600\" height=\"300\" src=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/09\/o-cubo-e-como-calcular-sua-\u00e1rea-total-e-volume.jpg\" alt=\"Como calcular volume e \u00e1rea total do Cubo?\" class=\"wp-image-18132\" srcset=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/09\/o-cubo-e-como-calcular-sua-\u00e1rea-total-e-volume.jpg 600w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/09\/o-cubo-e-como-calcular-sua-\u00e1rea-total-e-volume-300x150.jpg 300w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/09\/o-cubo-e-como-calcular-sua-\u00e1rea-total-e-volume-320x160.jpg 320w\" sizes=\"(max-width: 600px) 100vw, 600px\" \/><\/figure><\/div>\n\n\n<p>O <strong>volume<\/strong> <strong>\u00e9 dado multiplicando os valores das tr\u00eas dimens\u00f5es<\/strong>. Como o <strong>cubo \u00e9 formado<\/strong> por <strong>arestas de medidas iguais<\/strong>, podemos calcular seu volume <strong>simplesmente como:<\/strong><\/p>\n\n\n\n<ul><li>V = (aresta)\u00b3<\/li><\/ul>\n\n\n\n<p>Da mesma forma, o <strong>cubo possui 6 faces de \u00e1reas iguais<\/strong>, portanto, a sua <strong>\u00e1rea total<\/strong> pode ser dada <strong>simplesmente por:<\/strong><\/p>\n\n\n\n<ul><li>At = 6 . (aresta)\u00b2<\/li><\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Paralelep\u00edpedo volume e \u00e1rea total<\/h3>\n\n\n\n<figure class=\"wp-block-gallery aligncenter has-nested-images columns-default is-cropped wp-block-gallery-1 is-layout-flex\">\n<figure class=\"wp-block-image\"><img decoding=\"async\" loading=\"lazy\" width=\"600\" height=\"300\" data-id=\"18131\"  src=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/09\/o-paralelepipedo-e-como-calcular-sua-\u00e1rea-total-e-volume.jpg\" alt=\"Como calcular volume e \u00e1rea total do Paralelep\u00edpedo?\" class=\"wp-image-18131\" srcset=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/09\/o-paralelepipedo-e-como-calcular-sua-\u00e1rea-total-e-volume.jpg 600w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/09\/o-paralelepipedo-e-como-calcular-sua-\u00e1rea-total-e-volume-300x150.jpg 300w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/09\/o-paralelepipedo-e-como-calcular-sua-\u00e1rea-total-e-volume-320x160.jpg 320w\" sizes=\"(max-width: 600px) 100vw, 600px\" \/><\/figure>\n<\/figure>\n\n\n\n<p>Se um paralelep\u00edpedo possui dimens\u00f5es a, b, c; ent\u00e3o:<\/p>\n\n\n\n<ul><li>V = a . b . c<\/li><li>At = 2ab + 2ac + 2bc<\/li><\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Prisma e Pir\u00e2mide: volume e \u00e1rea total<\/h3>\n\n\n\n<p>O volume e a \u00e1rea total do prisma e da pir\u00e2mide <strong>dependem do pol\u00edgono que est\u00e1 na base<\/strong>, por isso usamos <strong>Ab: \u00e1rea da base <\/strong>(pode ser um tri\u00e2ngulo, quadrado, hex\u00e1gono, etc.) e <strong>Al: \u00e1rea lateral<\/strong> (ret\u00e2ngulos para prismas e tri\u00e2ngulos para pir\u00e2mides).&nbsp;<\/p>\n\n\n\n<p>O <strong>volume<\/strong> deles <strong>dependem diretamente da \u00e1rea da base e da altura (h)<\/strong>!<\/p>\n\n\n\n<p>Exemplo: <strong>Prisma <\/strong>de base<strong> triangular <\/strong>(pbt)<\/p>\n\n\n\n<figure class=\"wp-block-gallery aligncenter has-nested-images columns-default is-cropped wp-block-gallery-3 is-layout-flex\">\n<figure class=\"wp-block-image\"><img decoding=\"async\" loading=\"lazy\" width=\"600\" height=\"300\" data-id=\"18130\"  src=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/09\/o-prisma-de-base-triangular-e-como-calcular-sua-\u00e1rea-total-e-volume.jpg\" alt=\"Como calcular volume e \u00e1rea total do Prisma?\" class=\"wp-image-18130\" srcset=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/09\/o-prisma-de-base-triangular-e-como-calcular-sua-\u00e1rea-total-e-volume.jpg 600w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/09\/o-prisma-de-base-triangular-e-como-calcular-sua-\u00e1rea-total-e-volume-300x150.jpg 300w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/09\/o-prisma-de-base-triangular-e-como-calcular-sua-\u00e1rea-total-e-volume-320x160.jpg 320w\" sizes=\"(max-width: 600px) 100vw, 600px\" \/><\/figure>\n<\/figure>\n\n\n\n<ul><li>V (pbt) = Ab . h&nbsp;<\/li><li>At (pbt) = 2Ab + Al&nbsp;<\/li><\/ul>\n\n\n\n<p>(se as 2 bases s\u00e3o um tri\u00e2ngulo, logo, ab ser\u00e1 a \u00e1rea dos 2&nbsp; tri\u00e2ngulos!)&nbsp;<\/p>\n\n\n\n<p>Exemplo: Pir\u00e2mide de base quadrada (pbq)<\/p>\n\n\n\n<figure class=\"wp-block-gallery aligncenter has-nested-images columns-default is-cropped wp-block-gallery-5 is-layout-flex\">\n<figure class=\"wp-block-image\"><img decoding=\"async\" loading=\"lazy\" width=\"600\" height=\"300\" data-id=\"18129\"  src=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/09\/a-piramede-de-base-quadrada-e-como-calcular-sua-\u00e1rea-total-e-volume.jpg\" alt=\"Como calcular volume e \u00e1rea total da Pir\u00e2mide?\" class=\"wp-image-18129\" srcset=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/09\/a-piramede-de-base-quadrada-e-como-calcular-sua-\u00e1rea-total-e-volume.jpg 600w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/09\/a-piramede-de-base-quadrada-e-como-calcular-sua-\u00e1rea-total-e-volume-300x150.jpg 300w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/09\/a-piramede-de-base-quadrada-e-como-calcular-sua-\u00e1rea-total-e-volume-320x160.jpg 320w\" sizes=\"(max-width: 600px) 100vw, 600px\" \/><\/figure>\n<\/figure>\n\n\n\n<ul><li>V (pbt) = \u2153 Ab . h&nbsp;<\/li><\/ul>\n\n\n\n<p>(diferente dos primas, a pir\u00e2mide sofre um <strong>afunilamento no volume<\/strong>, por isso, <strong>contabilizamos apenas \u2153 <\/strong>da \u00e1rea da base).&nbsp;<\/p>\n\n\n\n<ul><li>At (pbt) = Ab + Al&nbsp;<\/li><\/ul>\n\n\n\n<p>(se a base \u00e9 um quadrado, ab ser\u00e1 a f\u00f3rmula da \u00e1rea do quadrado)&nbsp;<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Cilindro: volume e \u00e1rea total<\/h3>\n\n\n\n<figure class=\"wp-block-gallery aligncenter has-nested-images columns-default is-cropped wp-block-gallery-7 is-layout-flex\">\n<figure class=\"wp-block-image\"><img decoding=\"async\" loading=\"lazy\" width=\"600\" height=\"300\" data-id=\"18128\"  src=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/09\/o-cilindro-e-como-calcular-sua-\u00e1rea-total-e-volume.jpg\" alt=\"Como calcular volume e \u00e1rea total do Cilindro??\" class=\"wp-image-18128\" srcset=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/09\/o-cilindro-e-como-calcular-sua-\u00e1rea-total-e-volume.jpg 600w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/09\/o-cilindro-e-como-calcular-sua-\u00e1rea-total-e-volume-300x150.jpg 300w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/09\/o-cilindro-e-como-calcular-sua-\u00e1rea-total-e-volume-320x160.jpg 320w\" sizes=\"(max-width: 600px) 100vw, 600px\" \/><\/figure>\n<\/figure>\n\n\n\n<p>As medidas de corpos redondos s\u00e3o dadas pelo <strong>raio (r) e altura (h). <\/strong>Portanto:<\/p>\n\n\n\n<ul><li>V = \u03c0 .r . 2 . h<\/li><li>At = 2. \u03c0. r . (r + h)<\/li><\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Cone: volume e \u00e1rea total<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" loading=\"lazy\" width=\"600\" height=\"300\" src=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/09\/o-cone-e-como-calcular-sua-\u00e1rea-total-e-volume.jpg\" alt=\"Como calcular  volume e \u00e1rea total do cone?\" class=\"wp-image-18127\" srcset=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/09\/o-cone-e-como-calcular-sua-\u00e1rea-total-e-volume.jpg 600w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/09\/o-cone-e-como-calcular-sua-\u00e1rea-total-e-volume-300x150.jpg 300w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/09\/o-cone-e-como-calcular-sua-\u00e1rea-total-e-volume-320x160.jpg 320w\" sizes=\"(max-width: 600px) 100vw, 600px\" \/><\/figure><\/div>\n\n\n<p>Para calcular a medida da \u00e1rea do cone, primeiro <strong>precisamos descobrir a geratriz (g),<\/strong> segmentos que iniciam na extremidade do v\u00e9rtice, percorre a superf\u00edcie do cone e termina em um v\u00e9rtice da circunfer\u00eancia.<\/p>\n\n\n\n<p>Para calcular a geratriz em um cone reto, basta utilizar <a rel=\"noreferrer noopener\" aria-label=\" (abre numa nova aba)\" href=\"https:\/\/beduka.com\/blog\/materias\/matematica\/trigonometria-no-triangulo-retangulo\/\" target=\"_blank\">trigonometria<\/a> ou o <a href=\"https:\/\/beduka.com\/blog\/exercicios\/matematica-exercicios\/exercicios-sobre-o-teorema-de-pitagoras\/\" target=\"_blank\" rel=\"noreferrer noopener\" aria-label=\" (abre numa nova aba)\">teorema de pit\u00e1goras<\/a>!<\/p>\n\n\n\n<ul><li>At = \u03c0. r . (g + r)<\/li><li>V = \u2153 . \u03c0 . r . 2 . h&nbsp;<\/li><\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Esfera: volume e \u00e1rea total<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" loading=\"lazy\" width=\"600\" height=\"300\" src=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/09\/a-esfera-e-como-calcular-sua-\u00e1rea-total-e-volume.jpg\" alt=\"Como calcular volume e \u00e1rea total da Esfera?\" class=\"wp-image-18126\" srcset=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/09\/a-esfera-e-como-calcular-sua-\u00e1rea-total-e-volume.jpg 600w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/09\/a-esfera-e-como-calcular-sua-\u00e1rea-total-e-volume-300x150.jpg 300w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/09\/a-esfera-e-como-calcular-sua-\u00e1rea-total-e-volume-320x160.jpg 320w\" sizes=\"(max-width: 600px) 100vw, 600px\" \/><\/figure><\/div>\n\n\n<p>Esse \u00e9 o mais diferente de todos pois n\u00e3o possui faces com planos definidos!<\/p>\n\n\n\n<ul><li>At = 4 . \u03c0 . r\u00b2<\/li><li>V = 4 \u2044 3. \u03c0 . r\u00b3<\/li><\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">O que \u00e9 a Rela\u00e7\u00e3o de Euler?<\/h3>\n\n\n\n<p><strong>O matem\u00e1tico Euler percebeu uma rela\u00e7\u00e3o entre o n\u00famero de v\u00e9rtices (V), arestas (A) e faces (F)<\/strong>, conhecida como rela\u00e7\u00e3o de Euler, dada pela express\u00e3o:<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>V \u2013 A + F = 2<\/strong><\/p>\n\n\n\n<p><strong>Leia-se: <\/strong>v\u00e9rtices menos arestas mais faces \u00e9 igual a dois<\/p>\n\n\n\n<p>Logo, com base na equa\u00e7\u00e3o, descobre-se a quantidade de arestas que um s\u00f3lido possui pelo n\u00famero de faces e de v\u00e9rtices.<\/p>\n\n\n\n<ul><li>Aproveite a oportunidade para <a href=\"https:\/\/beduka.com\/blog\/exercicios\/matematica-exercicios\/exercicios-sobre-geometria-plana\/\" target=\"_blank\" rel=\"noreferrer noopener\" aria-label=\" (abre numa nova aba)\">aprender Geometria Plana!<\/a><\/li><\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">Exerc\u00edcios de Geometria Espacial com Gabarito<\/h2>\n\n\n\n<p>No in\u00edcio, essa mat\u00e9ria at\u00e9 assusta, mas agora sei que com esse resumo tudo ficou mais claro para voc\u00ea. Que tal testar seu conhecimento com <strong>Exerc\u00edcios de Fun\u00e7\u00e3o Exponencial.<\/strong>&nbsp;<\/p>\n\n\n\n<p><strong>Parab\u00e9ns<\/strong> por ter lido at\u00e9 aqui!!!&nbsp;<\/p>\n\n\n\n<p>Baixe <strong>gratuitamente<\/strong> o <a href=\"https:\/\/beduka.com\/blog\/dicas\/enem-dicas\/plano-de-estudos-gratuito\/\" target=\"_blank\" rel=\"noreferrer noopener\" aria-label=\" (abre numa nova aba)\">Plano de Estudos<\/a> do <strong>Beduka<\/strong> e tenha uma prepara\u00e7\u00e3o<strong> perfeita <\/strong>para o <strong>ENEM<\/strong>.<\/p>\n\n\n\n<p>Quest\u00e3o 1 &#8211; <a href=\"https:\/\/enem.inep.gov.br\/\" target=\"_blank\" rel=\"noreferrer noopener\" aria-label=\" (abre numa nova aba)\">(Enem)<\/a> Para resolver o problema de abastecimento de \u00e1gua, foi decidida, numa reuni\u00e3o do condom\u00ednio, a constru\u00e7\u00e3o de uma nova cisterna. A cisterna atual tem formato cil\u00edndrico, com 3 m de altura e 2 m de di\u00e2metro, e estimou-se que a nova cisterna dever\u00e1 comportar 81 m\u00b3 de \u00e1gua, mantendo o formato cil\u00edndrico e a altura da atual. Ap\u00f3s a inaugura\u00e7\u00e3o da nova cisterna, a antiga ser\u00e1 desativada. (Utilize 3,0 como aproxima\u00e7\u00e3o para \u03c0.)<\/p>\n\n\n\n<p>Qual deve ser o aumento, em metros, no raio da cisterna para atingir o volume desejado?<\/p>\n\n\n\n<p><strong>a)<\/strong> 0,5<\/p>\n\n\n\n<p><strong>b)<\/strong> 1,0<\/p>\n\n\n\n<p><strong>c)<\/strong> 2,0<\/p>\n\n\n\n<p><strong>d)<\/strong> 3,5<\/p>\n\n\n\n<p><strong>e)<\/strong> 8,0<\/p>\n\n\n\n<p><strong>Quest\u00e3o 2 &#8211;<\/strong> <a href=\"https:\/\/www.ifg.edu.br\/\" target=\"_blank\" rel=\"noreferrer noopener\" aria-label=\" (abre numa nova aba)\">(IFG)<\/a> As medidas internas de um reservat\u00f3rio no formato de um paralelep\u00edpedo s\u00e3o de 2,5 m de comprimento, 1,8 m de largura e 1,2 m de profundidade (altura). Se, em um determinado momento do dia, esse reservat\u00f3rio est\u00e1 apenas com 70% de sua capacidade, a quantidade de litros que faltam para ench\u00ea-lo \u00e9 igual a:<\/p>\n\n\n\n<p><strong>a) <\/strong>1620<\/p>\n\n\n\n<p><strong>b) <\/strong>1630<\/p>\n\n\n\n<p><strong>c) <\/strong>1640<\/p>\n\n\n\n<p><strong>d) <\/strong>1650<\/p>\n\n\n\n<p><strong>e) <\/strong>1660<\/p>\n\n\n\n<p><strong>Quest\u00e3o 3 &#8211;<\/strong> <a href=\"http:\/\/sistemasenem2.inep.gov.br\/\" target=\"_blank\" rel=\"noreferrer noopener\" aria-label=\" (abre numa nova aba)\">(Enem 2010)<\/a> A sider\u00fargica \u201cMetal Nobre\u201d produz diversos objetos maci\u00e7os utilizando o ferro. Um tipo especial de pe\u00e7a feita nessa companhia tem o formato de um paralelep\u00edpedo retangular, de acordo com as dimens\u00f5es indicadas na figura que segue.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/X8Ot_6L_iBPKImfHW08MJxhip1mFc55ySfODedaNY8scbZ6s5Jpr2kXj3xqqh_Wl7PNd69nIUGgoGps8r0xTgSG1wptxkrHeN9bLgMJueOU1CaiWnCqICkLmifq8B6IGnzx6Nw3U\" alt=\"questaoenem155\"\/><\/figure><\/div>\n\n\n<p>O produto das tr\u00eas dimens\u00f5es indicadas na pe\u00e7a resultaria na medida da grandeza:<\/p>\n\n\n\n<p><strong>a) <\/strong>massa<\/p>\n\n\n\n<p><strong>b) <\/strong>volume<\/p>\n\n\n\n<p><strong>c) <\/strong>superf\u00edcie<\/p>\n\n\n\n<p><strong>d) <\/strong>capacidade<\/p>\n\n\n\n<p><strong>e) <\/strong>comprimento<\/p>\n\n\n\n<p><strong>Quest\u00e3o 4 &#8211; <\/strong><a href=\"http:\/\/www.ufrgs.br\/coperse\/concurso-vestibular\/anteriores\/2017\" target=\"_blank\" rel=\"noreferrer noopener\" aria-label=\" (abre numa nova aba)\">(UFRGS 2017)<\/a> Considere a planifica\u00e7\u00e3o de um tetraedro, conforme a figura abaixo.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/yYzyOFCvyp_hFit_c944R0hGITOQk2KsxZZIxrri5RCkSBbKOYbWSHrjc1p5j26H2GnnZV3C6JpJ66FQaCd17THPKaTCwod9aG-t7-N35ACC1QSRauqNFecZJQvz3pSI_vadEh4V\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>Os tri\u00e2ngulos ABC e ABD s\u00e3o is\u00f3sceles respectivamente em B e D. As medidas dos segmentos <em>AC<\/em> ,<em> BC<\/em> , <em>BD<\/em> e <em>DF<\/em> est\u00e3o indicadas na figura.<\/p>\n\n\n\n<p>A soma das medidas de todas as arestas do tetraedro \u00e9:<\/p>\n\n\n\n<p><strong>a) <\/strong>33<\/p>\n\n\n\n<p><strong>b) <\/strong>34<\/p>\n\n\n\n<p><strong>c) <\/strong>43<\/p>\n\n\n\n<p><strong>d) <\/strong>47<\/p>\n\n\n\n<p><strong>e) <\/strong>48<\/p>\n\n\n\n<p><strong>Quest\u00e3o 5 &#8211;<\/strong> <a href=\"https:\/\/www.fuvest.br\/\" target=\"_blank\" rel=\"noreferrer noopener\" aria-label=\" (abre numa nova aba)\">(Fuvest-SP)<\/a> Uma superf\u00edcie esf\u00e9rica de raio 13 cm \u00e9 cortada por um plano situado a uma dist\u00e2ncia de 12 cm do centro da superf\u00edcie esf\u00e9rica, determinando uma circunfer\u00eancia. O raio dessa circunfer\u00eancia, em cent\u00edmetros, \u00e9<\/p>\n\n\n\n<p><strong>a)<\/strong> &nbsp; 5&nbsp;&nbsp;&nbsp;<\/p>\n\n\n\n<p><strong>b)<\/strong> &nbsp; 4<\/p>\n\n\n\n<p><strong>c)<\/strong> &nbsp; 3<\/p>\n\n\n\n<p><strong>d)<\/strong> &nbsp; 2<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Gabarito dos Exerc\u00edcios de Geometria Espacial<\/h3>\n\n\n\n<p><strong>Exerc\u00edcio resolvido da quest\u00e3o 1 &#8211;<\/strong><\/p>\n\n\n\n<p>Alternativa correta:  <strong>c)<\/strong> 2,0 <\/p>\n\n\n\n<p><strong>Exerc\u00edcio resolvido da quest\u00e3o 2 &#8211;<\/strong><\/p>\n\n\n\n<p>Alternativa correta: <strong>a) <\/strong>1620<\/p>\n\n\n\n<p><strong>Exerc\u00edcio resolvido da quest\u00e3o 3 &#8211;<\/strong><\/p>\n\n\n\n<p>Alternativa correta: <strong>b)<\/strong> volume<\/p>\n\n\n\n<p><strong>Exerc\u00edcio resolvido da quest\u00e3o 4 &#8211;<\/strong><\/p>\n\n\n\n<p>Alternativa correta: <strong>a)<\/strong> 33<\/p>\n\n\n\n<p><strong>Exerc\u00edcio resolvido da quest\u00e3o 5 &#8211;<\/strong><\/p>\n\n\n\n<p>Alternativa correta: <strong>a)<\/strong> &nbsp; 5&nbsp;&nbsp;<\/p>\n\n\n\n<p>Estude para o Enem com o <a href=\"https:\/\/beduka.com\/simulado\" target=\"_blank\" rel=\"noreferrer noopener\" aria-label=\" (abre numa nova aba)\">Simulado Beduka.<\/a> <strong>\u00c9 gratuito!<\/strong><\/p>\n\n\n\n<p>Gostou dos nossos <strong>Exerc\u00edcios de Geometria Espacial?<\/strong> Compartilhe com os seus amigos e comente abaixo sobre as \u00e1reas que voc\u00ea deseja mais explica\u00e7\u00f5es.<\/p>\n\n\n\n<p>Queremos te ajudar a encontrar a <strong>FACULDADE IDEAL! <\/strong>Logo abaixo, fa\u00e7a uma pesquisa por curso e cidade que te mostraremos todas as faculdades que podem te atender. <strong>Informamos a nota de corte, valor de mensalidade, nota do MEC, avalia\u00e7\u00e3o dos alunos, modalidades de ensino e muito mais.<\/strong><br><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Geometria espacial \u00e9 a \u00e1rea da Matem\u00e1tica respons\u00e1vel pela investiga\u00e7\u00e3o de s\u00f3lidos no espa\u00e7o, ou seja, ela analisa objetos tridimensionais. Essas formas s\u00e3o compostas por pontos, retas e planos. Leia nosso resumo e coloque em pr\u00e1tica todo o seu conhecimento com Exerc\u00edcios de Geometria Espacial. Quando voc\u00ea terminar os Exerc\u00edcios de Geometria Espacial, fa\u00e7a nosso [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":18139,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_mi_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0},"categories":[610],"tags":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v20.10 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Exerc\u00edcios de Geometria Espacial com Gabarito<\/title>\n<meta name=\"description\" content=\"Leia nosso resumo e coloque seu conhecimento em pr\u00e1tica com uma Lista de Exerc\u00edcios de Geometria Espacial selecionados para voc\u00ea.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" 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