{"id":17380,"date":"2020-08-12T08:51:11","date_gmt":"2020-08-12T11:51:11","guid":{"rendered":"https:\/\/beduka.com\/blog\/?p=17380"},"modified":"2021-06-25T11:31:01","modified_gmt":"2021-06-25T14:31:01","slug":"geometria-espacial","status":"publish","type":"post","link":"https:\/\/beduka.com\/blog\/materias\/matematica\/geometria-espacial\/","title":{"rendered":"Entenda a Geometria Espacial: principais conceitos e f\u00f3rmulas!"},"content":{"rendered":"\n<blockquote class=\"wp-block-quote\"><p><strong><em>A Geometria Espacial \u00e9 a \u00e1rea que estuda os s\u00f3lidos geom\u00e9tricos, ou seja, aqueles que possuem tr\u00eas dimens\u00f5es e n\u00e3o &#8220;cabem&#8221; no papel. Al\u00e9m de calcular a \u00e1rea, aprendemos uma novidade: como encontrar o volume. Neste resumo, voc\u00ea aprende tudo o que \u00e9 importante saber:<\/em><\/strong><em><strong> defini\u00e7\u00f5es, exemplos e f\u00f3rmulas.<\/strong><\/em><br><\/p><\/blockquote>\n\n\n\n<p>Neste artigo sobre Geometria Espacial voc\u00ea encontrar\u00e1 todos os t\u00f3picos abaixo. <strong>Clique em um deles para ir direto ao assunto!<\/strong><\/p>\n\n\n\n<ol><li><a href=\"#1\">O que \u00e9 Geometria Espacial?<\/a><\/li><li><a href=\"#2\">Quais s\u00e3o as formas geom\u00e9tricas espaciais e seus tipos?<\/a><\/li><li><a href=\"#3\">A importante Rela\u00e7\u00e3o de Euler!<\/a><\/li><li><a href=\"#4\">Para que serve o estudo da geometria espacial?<\/a><\/li><li><a href=\"#5\">Como calcular \u00e1rea total e volume dos s\u00f3lidos?<\/a><\/li><\/ol>\n\n\n\n<ul><li>Estudando para as provas? Conhe\u00e7a nosso <a href=\"https:\/\/beduka.com\/blog\/dicas\/enem-dicas\/simulados-para-enem-online-e-gratuito\/\" target=\"_blank\" rel=\"noreferrer noopener\" aria-label=\" (abre numa nova aba)\">Simulado gratuito<\/a>, que pode ser personalizado com as mat\u00e9rias que voc\u00ea mais precisa!<\/li><\/ul>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"1\">O que \u00e9 Geometria Espacial?<\/h2>\n\n\n\n<p>A Geometria Espacial \u00e9 a \u00e1rea da matem\u00e1tica que <strong>estuda os s\u00f3lidos geom\u00e9tricos<\/strong>. Esses s\u00e3o as formas que constru\u00edmos com <strong>3 dimens\u00f5es: altura, largura e profundidade.<\/strong> <\/p>\n\n\n\n<p>Por isso, surge a novidade de calcular <strong>n\u00e3o s\u00f3 a \u00e1rea total, mas tamb\u00e9m o volume<\/strong> do s\u00f3lido.<\/p>\n\n\n\n<p>Voc\u00ea j\u00e1 deve ter ouvido falar em <a rel=\"noreferrer noopener\" aria-label=\" (abre numa nova aba)\" href=\"https:\/\/beduka.com\/blog\/exercicios\/matematica-exercicios\/exercicios-sobre-geometria-plana\/\" target=\"_blank\">Geometria Plana<\/a>, que estuda as figuras 2D (como uma folha de papel, tem largura e altura). <strong>Apesar de diferentes, \u00e9  importante compreend\u00ea-la para depois avan\u00e7armos<\/strong> \u00e0 Geometria Espacial.<\/p>\n\n\n\n<p>Os<strong> conceitos b\u00e1sicos <\/strong>de qualquer geometria s\u00e3o os mesmos: reta, ponto, plano. Vamos relembr\u00e1-los:<\/p>\n\n\n\n<ul><li><strong>Ponto:<\/strong> \u00e9 aquilo que n\u00e3o tem partes, indivis\u00edvel. Ele marca posi\u00e7\u00e3o no espa\u00e7o, junto \u00e0s coordenadas.<\/li><li><strong>Reta:<\/strong> \u00e9 um segmento infinito composto por pontos lineares, tamb\u00e9m \u00e9 definida como a dist\u00e2ncia entre, no m\u00ednimo, 2 pontos.<\/li><li><strong>Linha:<\/strong> \u00e9 semelhante \u00e0 reta, mas tem a propriedade de formar curvas e n\u00f3s sobre si mesma.<\/li><li><strong>Plano:<\/strong> \u00e9 uma estrutura infinita que se estende em todas as dire\u00e7\u00f5es, e cada plano determina uma medida: a altura pertence a um plano, o comprimento a outro e a profundidade a outro.<\/li><li><strong>Espa\u00e7o:<\/strong> \u00e9 a uni\u00e3o dos 3 planos, o que garante o \u201cefeito 3d\u201d.<\/li><\/ul>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" loading=\"lazy\" width=\"600\" height=\"300\" src=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/08\/o-que-e-ponto-reta-planos-geometricos-e-as-tres-dimensoes-dos-s\u00f3lidos-largura-altura-e-profundidade.jpg\" alt=\"o-que-e-ponto-reta-planos-geometricos-e-as-tres-dimensoes-dos-s\u00f3lidos-largura-altura-e-profundidade\" class=\"wp-image-17381\" srcset=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/08\/o-que-e-ponto-reta-planos-geometricos-e-as-tres-dimensoes-dos-s\u00f3lidos-largura-altura-e-profundidade.jpg 600w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/08\/o-que-e-ponto-reta-planos-geometricos-e-as-tres-dimensoes-dos-s\u00f3lidos-largura-altura-e-profundidade-300x150.jpg 300w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/08\/o-que-e-ponto-reta-planos-geometricos-e-as-tres-dimensoes-dos-s\u00f3lidos-largura-altura-e-profundidade-320x160.jpg 320w\" sizes=\"(max-width: 600px) 100vw, 600px\" \/><\/figure><\/div>\n\n\n\n<p>Ainda \u00e9 importante relembrar alguns<strong> termos e rela\u00e7\u00f5es <\/strong>entre esses elementos, porque eles podem <strong>aparecer em quest\u00f5es e explica\u00e7\u00f5es<\/strong>:<\/p>\n\n\n\n<ul><li>Os pontos ainda podem ser<strong> coplanares<\/strong> quando pertencem ao mesmo plano e <strong>colineares <\/strong>quando pertencem a uma mesma linha.<\/li><li>Quando as retas pertencem ao mesmo plano, elas podem ser <strong>paralelas, concorrentes ou coincidentes<\/strong>. Contudo, se n\u00e3o forem coplanares, elas ser\u00e3o chamadas de <strong>reversas<\/strong>.<\/li><li>J\u00e1 os planos, quando s\u00e3o paralelos, n\u00e3o compartilham nenhum ponto em comum. Quando eles se interceptam de alguma forma, s\u00e3o chamados de <strong>secantes,<\/strong> e quando h\u00e1 dois ocupando o mesmo espa\u00e7o, s\u00e3o chamados de coincidentes.<\/li><li>Por \u00faltimo, tamb\u00e9m podemos relacionar retas e planos, lembrando que elas podem <strong>pertencer <\/strong>ao plano, serem <strong>secantes <\/strong>(apenas toc\u00e1-lo em um ponto) ou serem<strong> paralelas <\/strong>(quando n\u00e3o possuem nada em comum).<\/li><\/ul>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"2\">Quais s\u00e3o as formas geom\u00e9tricas espaciais e seus tipos?<\/h2>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" loading=\"lazy\" width=\"600\" height=\"300\" src=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/08\/O-cubo-e-sesu-elementos-aresta-face-e-v\u00e9rtice.jpg\" alt=\"O-cubo-e-sesu-elementos-aresta-face-e-v\u00e9rtice\" class=\"wp-image-17383\" srcset=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/08\/O-cubo-e-sesu-elementos-aresta-face-e-v\u00e9rtice.jpg 600w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/08\/O-cubo-e-sesu-elementos-aresta-face-e-v\u00e9rtice-300x150.jpg 300w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/08\/O-cubo-e-sesu-elementos-aresta-face-e-v\u00e9rtice-320x160.jpg 320w\" sizes=\"(max-width: 600px) 100vw, 600px\" \/><\/figure><\/div>\n\n\n\n<p>Como j\u00e1 foi dito, os s\u00f3lidos geom\u00e9tricos s\u00e3o aqueles que possuem tr\u00eas dimens\u00f5es. Al\u00e9m disso, devem ter <strong>pelo menos um dos tr\u00eas elementos<\/strong> seguintes: <strong>v\u00e9rtice, aresta e face.<\/strong><\/p>\n\n\n\n<p>Eles tamb\u00e9m podem ser classificados em <strong>tipos:<\/strong><\/p>\n\n\n\n<ul><li><strong>Poliedros<\/strong>: s\u00e3o fechados e constitu\u00eddos de faces poligonais, al\u00e9m de ter os 3 elementos (v\u00e9rtice, aresta e face). <strong>Exemplos: <\/strong>cubos, paralelep\u00edpedos, prismas, pir\u00e2mides, etc.<\/li><li><strong>Corpos redondos<\/strong>: tamb\u00e9m conhecidos como s\u00f3lidos de revolu\u00e7\u00e3o, possuem esse nome porque s\u00e3o criados ao tomar um plano e gir\u00e1-lo, vendo qual figura forma em movimento. S\u00e3o aqueles que apresentam curvas e a sua medida espec\u00edfica \u00e9 o<strong> raio (r)<\/strong>. <strong>Exemplos:<\/strong> <a href=\"https:\/\/beduka.com\/blog\/materias\/resumo-de-cilindro\/\" data-type=\"post\" data-id=\"21849\" target=\"_blank\" rel=\"noreferrer noopener\">cilindro<\/a>, cone, esfera (rota\u00e7\u00e3o do c\u00edrculo), etc.<\/li><li><strong>S\u00f3lidos de Plat\u00e3o<\/strong>: s\u00e3o casos particulares de poliedros regulares, convexos, e congruentes que <a rel=\"noreferrer noopener\" aria-label=\" (abre numa nova aba)\" href=\"https:\/\/beduka.com\/blog\/materias\/filosofia\/principais-ideias-platao\/\" target=\"_blank\">Plat\u00e3o<\/a> teorizou e relacionou com os elementos da natureza. <strong>Os 5 s\u00f3lidos cl\u00e1ssicos s\u00e3o<\/strong>: tetraedro (pir\u00e2mide), hexaedro (cubo), octaedro (8 faces), dodecaedro (12 faces) e icosaedro (20 lados).<\/li><\/ul>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" loading=\"lazy\" width=\"600\" height=\"300\" src=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/08\/tipos-de-s\u00f3lidos-geom\u00e9tricos-s\u00f3lidos-de-plat\u00e3o.jpg\" alt=\"tipos-de-s\u00f3lidos-geom\u00e9tricos-s\u00f3lidos-de-plat\u00e3o e corpos redondos\" class=\"wp-image-17384\" srcset=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/08\/tipos-de-s\u00f3lidos-geom\u00e9tricos-s\u00f3lidos-de-plat\u00e3o.jpg 600w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/08\/tipos-de-s\u00f3lidos-geom\u00e9tricos-s\u00f3lidos-de-plat\u00e3o-300x150.jpg 300w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/08\/tipos-de-s\u00f3lidos-geom\u00e9tricos-s\u00f3lidos-de-plat\u00e3o-320x160.jpg 320w\" sizes=\"(max-width: 600px) 100vw, 600px\" \/><\/figure><\/div>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"3\">O que \u00e9 Rela\u00e7\u00e3o de Euler?<\/h2>\n\n\n\n<p>Um matem\u00e1tico chamado Euler percebeu que <strong>no poliedro havia uma rela\u00e7\u00e3o constante <\/strong>\u00a0entre o n\u00famero de v\u00e9rtices (V), faces (F) e arestas (A). <\/p>\n\n\n\n<p><strong>A f\u00f3rmula que descreve esse padr\u00e3o \u00e9 chamada de rela\u00e7\u00e3o de Euler<\/strong>, e \u00e9 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 class=\"has-text-align-left\">Com essa rela\u00e7\u00e3o \u00e9 poss\u00edvel descobrir a quantidade de arestas que\u00a0 qualquer poliedro possui, se tivermos o n\u00famero de faces e de v\u00e9rtices.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Exemplo<\/strong><\/h3>\n\n\n\n<blockquote class=\"wp-block-quote\"><p><strong>1)<\/strong> Maria disse que montaria a maquete de um poliedro com 20 lados, e para isso separou 30 palitos para servir de arestas. Ela ainda pegou 10 bolinhas de isopor para formar os v\u00e9rtices. Ser\u00e1 poss\u00edvel montar este poliedro? Se n\u00e3o, quantas bolinhas de isopor faltariam?<\/p><\/blockquote>\n\n\n\n<p><strong>Solu\u00e7\u00e3o:<\/strong><\/p>\n\n\n\n<p>Primeiro, precisamos conferir se as quantidades que Maria reservou s\u00e3o suficientes para montar o poliedro. Faremos isso utilizando a rela\u00e7\u00e3o de Euler:<br><\/p>\n\n\n\n<p class=\"has-text-align-center\">V &#8211; A + F = 2<\/p>\n\n\n\n<p class=\"has-text-align-center\">10 &#8211; 30 + 20 = 2<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>0 = 2 (???)<\/strong><br><\/p>\n\n\n\n<p>Portanto, <strong>conclu\u00edmos que Maria n\u00e3o conseguiri<\/strong>a montar um poliedro com esses materiais.<\/p>\n\n\n\n<p>Agora, a quest\u00e3o pede qual \u00e9 o n\u00famero de v\u00e9rtices a mais para que ela conseguisse:<\/p>\n\n\n\n<p class=\"has-text-align-center\">V &#8211; A + F = 2<\/p>\n\n\n\n<p class=\"has-text-align-center\">V &#8211; 30 + 20 = 2<\/p>\n\n\n\n<p class=\"has-text-align-center\">V = 2 + 30 &#8211; 20<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>V = 12<\/strong><\/p>\n\n\n\n<p>Dessa forma,<strong> ser\u00e1 necess\u00e1rio que Maria<\/strong> <strong>arranje 2 bolinhas a mais,<\/strong> pois com 12 v\u00e9rtices a rela\u00e7\u00e3o se verifica verdadeira!<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"4\">Para que serve o estudo da Geometria Espacial?<\/h2>\n\n\n\n<p>Agora que voc\u00ea j\u00e1 entendeu do que se trata, deve ficar pensando qual \u00e9 o sentido de fazer contas com esses elementos!<\/p>\n\n\n\n<p>O estudo da Geometria Espacial serve como <strong>base do nosso dia a dia<\/strong>. Ele compreende as <strong>primeiras<\/strong> <strong>no\u00e7\u00f5es de espa\u00e7o e tamanho<\/strong>.<\/p>\n\n\n\n<p>Algumas situa\u00e7\u00f5es em que \u00e9 preciso ter a <a href=\"https:\/\/beduka.com\/blog\/materias\/matematica\/visao-espacial-matematica\/\" data-type=\"post\" data-id=\"18615\">vis\u00e3o espacial<\/a>, s\u00e3o:<\/p>\n\n\n\n<p>Quando um engenheiro constr\u00f3i um pr\u00e9dio, um arquiteto o esquematiza, ou at\u00e9 quando compramos uma bomba para encher a piscina.<\/p>\n\n\n\n<p>Al\u00e9m disso, <strong>na matem\u00e1tica da escola e nas provas,<\/strong> ela serve para lidar com f\u00f3rmulas e c\u00e1lculos das dimens\u00f5es dos s\u00f3lidos: \u00e1rea e volume. \u00c9 um <strong>assunto que costuma &#8220;cair&#8221; bastante!<\/strong><\/p>\n\n\n\n<ul><li><strong>Vamos aprender como fazer esses c\u00e1lculos:<\/strong><\/li><\/ul>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"5\">Como calcular \u00e1rea total e volume dos s\u00f3lidos?<\/h2>\n\n\n\n<p>Na Geometria Plana, o c\u00e1lculo do preenchimento da figura era a \u00e1rea, enquanto o do contorno era o per\u00edmetro.<\/p>\n\n\n\n<p>Na Geometria Espacial, n\u00f3s tamb\u00e9m calculamos. A diferen\u00e7a \u00e9 que, por ter mais dimens\u00f5es, o<strong> <\/strong>preenchimento ser\u00e1 o <strong>espa\u00e7o ocupado internamente (volume)<\/strong> e os contornos ser\u00e3o as<strong> \u00e1reas das faces (\u00e1rea total)<\/strong>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Cubo: volume e \u00e1rea total<\/h3>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" loading=\"lazy\" width=\"600\" height=\"300\" src=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/08\/o-cubo-e-como-calcular-sua-\u00e1rea-total-e-volume.jpg\" alt=\"o-cubo-e-como-calcular-sua-\u00e1rea-total-e-volume\" class=\"wp-image-17385\" srcset=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/08\/o-cubo-e-como-calcular-sua-\u00e1rea-total-e-volume.jpg 600w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/08\/o-cubo-e-como-calcular-sua-\u00e1rea-total-e-volume-300x150.jpg 300w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/08\/o-cubo-e-como-calcular-sua-\u00e1rea-total-e-volume-320x160.jpg 320w\" sizes=\"(max-width: 600px) 100vw, 600px\" \/><\/figure><\/div>\n\n\n\n<p>O cubo<strong> possui 6 faces de \u00e1reas iguais<\/strong>, portanto, a sua \u00e1rea total pode ser dada simplesmente por:<br><\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>At = 6 . (lado)\u00b2<\/strong><\/p>\n\n\n\n<p>O <strong>volume<\/strong> do cubo <strong>\u00e9 dado multiplicando os valores das tr\u00eas dimens\u00f5es<\/strong>. Como o cubo \u00e9 formado por <strong>arestas de medidas iguais<\/strong>, podemos calcular seu volume simplesmente como:<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>V = (aresta)\u00b3<\/strong><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Paralelep\u00edpedo volume e \u00e1rea total<\/h3>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" loading=\"lazy\" width=\"600\" height=\"300\" src=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/08\/o-paralelepipedo-e-como-calcular-sua-\u00e1rea-total-e-volume.jpg\" alt=\"o-paralelepipedo-e-como-calcular-sua-\u00e1rea-total-e-volume\" class=\"wp-image-17386\" srcset=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/08\/o-paralelepipedo-e-como-calcular-sua-\u00e1rea-total-e-volume.jpg 600w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/08\/o-paralelepipedo-e-como-calcular-sua-\u00e1rea-total-e-volume-300x150.jpg 300w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/08\/o-paralelepipedo-e-como-calcular-sua-\u00e1rea-total-e-volume-320x160.jpg 320w\" sizes=\"(max-width: 600px) 100vw, 600px\" \/><\/figure><\/div>\n\n\n\n<p>Se um paralelep\u00edpedo possui dimens\u00f5es a, b, c; ent\u00e3o:<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>At = 2ab + 2ac + 2bc<\/strong><\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>V = a . b . c<\/strong><\/p>\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>. <\/p>\n\n\n\n<p>Por isso usamos siglas para escrever a f\u00f3rmula geral e \u00e9 t\u00e3o importante ter aprendido a geometria plana antes. <\/p>\n\n\n\n<p><strong>Antes demostrar as f\u00f3rmulas, vamos fazer o combinado:<\/strong><\/p>\n\n\n\n<ul><li><strong>Ab = \u00e1rea da base<\/strong>: pode ser um tri\u00e2ngulo, quadrado, hex\u00e1gono, etc.<\/li><li><strong>Al = \u00e1rea lateral<\/strong>: s\u00e3o ret\u00e2ngulos nos prismas e tri\u00e2ngulos nas pir\u00e2mides.\u00a0<\/li><\/ul>\n\n\n\n<h4 class=\"wp-block-heading\">Exemplo: <strong>Prisma <\/strong>de base<strong> triangular <\/strong>(pbt)<br><\/h4>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" loading=\"lazy\" width=\"600\" height=\"300\" src=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/08\/o-prisma-de-base-triangular-e-como-calcular-sua-\u00e1rea-total-e-volume.jpg\" alt=\"o-prisma-de-base-triangular-e-como-calcular-sua-\u00e1rea-total-e-volume\" class=\"wp-image-17387\" srcset=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/08\/o-prisma-de-base-triangular-e-como-calcular-sua-\u00e1rea-total-e-volume.jpg 600w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/08\/o-prisma-de-base-triangular-e-como-calcular-sua-\u00e1rea-total-e-volume-300x150.jpg 300w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/08\/o-prisma-de-base-triangular-e-como-calcular-sua-\u00e1rea-total-e-volume-320x160.jpg 320w\" sizes=\"(max-width: 600px) 100vw, 600px\" \/><\/figure><\/div>\n\n\n\n<p class=\"has-text-align-center\"><strong>At (pbt) = 2Ab + Al\u00a0<\/strong><\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>V (pbt) = Ab . h<\/strong><\/p>\n\n\n\n<p>(se as 2 bases s\u00e3o um tri\u00e2ngulo, logo, ab ser\u00e1 a \u00e1rea dos 2 tri\u00e2ngulos!)\u00a0<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Exemplo:<strong> Pir\u00e2mide <\/strong>de base<strong> quadrada<\/strong> (pbq)<br><\/h4>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" loading=\"lazy\" width=\"600\" height=\"300\" src=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/08\/a-piramede-de-base-quadrada-e-como-calcular-sua-\u00e1rea-total-e-volume.jpg\" alt=\"a-piramede-de-base-quadrada-e-como-calcular-sua-\u00e1rea-total-e-volume\" class=\"wp-image-17388\" srcset=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/08\/a-piramede-de-base-quadrada-e-como-calcular-sua-\u00e1rea-total-e-volume.jpg 600w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/08\/a-piramede-de-base-quadrada-e-como-calcular-sua-\u00e1rea-total-e-volume-300x150.jpg 300w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/08\/a-piramede-de-base-quadrada-e-como-calcular-sua-\u00e1rea-total-e-volume-320x160.jpg 320w\" sizes=\"(max-width: 600px) 100vw, 600px\" \/><\/figure><\/div>\n\n\n\n<p class=\"has-text-align-center\"><strong>At (pbt) = Ab + Al<\/strong><\/p>\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<p class=\"has-text-align-center\"><strong>V (pbt) = \u2153 Ab . h<\/strong><\/p>\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).\u00a0<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Cilindro: volume e \u00e1rea total<\/h3>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" loading=\"lazy\" width=\"600\" height=\"300\" src=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/08\/o-cilindro-e-como-calcular-sua-\u00e1rea-total-e-volume.jpg\" alt=\"o-cilindro-e-como-calcular-sua-\u00e1rea-total-e-volume\" class=\"wp-image-17389\" srcset=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/08\/o-cilindro-e-como-calcular-sua-\u00e1rea-total-e-volume.jpg 600w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/08\/o-cilindro-e-como-calcular-sua-\u00e1rea-total-e-volume-300x150.jpg 300w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/08\/o-cilindro-e-como-calcular-sua-\u00e1rea-total-e-volume-320x160.jpg 320w\" sizes=\"(max-width: 600px) 100vw, 600px\" \/><\/figure><\/div>\n\n\n\n<p>As medidas de corpos redondos s\u00e3o dadas pelo <strong>raio (r) e altura (h). <\/strong>Portanto, o cilindro ser\u00e1:<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>At = 2. \u03c0. r . (r + h)<\/strong><\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>V = \u03c0 .r . 2 . <\/strong>h<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Cone: volume e \u00e1rea total<\/h3>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" loading=\"lazy\" width=\"600\" height=\"300\" src=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/08\/o-cone-e-como-calcular-sua-\u00e1rea-total-e-volume.jpg\" alt=\"o-cone-e-como-calcular-sua-\u00e1rea-total-e-volume\" class=\"wp-image-17390\" srcset=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/08\/o-cone-e-como-calcular-sua-\u00e1rea-total-e-volume.jpg 600w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/08\/o-cone-e-como-calcular-sua-\u00e1rea-total-e-volume-300x150.jpg 300w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/08\/o-cone-e-como-calcular-sua-\u00e1rea-total-e-volume-320x160.jpg 320w\" sizes=\"(max-width: 600px) 100vw, 600px\" \/><\/figure><\/div>\n\n\n\n<p>Para calcular a medida da \u00e1rea do cone <strong>precisamos, antes, descobrir a geratriz (g). <\/strong>Esse \u00e9 o segmento que inicia 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 rel=\"noreferrer noopener\" aria-label=\" (abre numa nova aba)\" href=\"https:\/\/beduka.com\/blog\/exercicios\/matematica-exercicios\/exercicios-sobre-o-teorema-de-pitagoras\/\" target=\"_blank\">teorema de pit\u00e1goras<\/a>!<\/p>\n\n\n\n<p>Depois, aplicamos os valor nas f\u00f3rmulas do Cone:<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>At = \u03c0.r.(g + r)<\/strong><\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong><strong>V= 1\/3 h.\u03c0.r\u00b2<\/strong><\/strong><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Esfera: volume e \u00e1rea total<\/h3>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" loading=\"lazy\" width=\"600\" height=\"300\" src=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/08\/a-esfera-e-como-calcular-sua-\u00e1rea-total-e-volume.jpg\" alt=\"a-esfera-e-como-calcular-sua-\u00e1rea-total-e-volume\" class=\"wp-image-17391\" srcset=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/08\/a-esfera-e-como-calcular-sua-\u00e1rea-total-e-volume.jpg 600w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/08\/a-esfera-e-como-calcular-sua-\u00e1rea-total-e-volume-300x150.jpg 300w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/08\/a-esfera-e-como-calcular-sua-\u00e1rea-total-e-volume-320x160.jpg 320w\" sizes=\"(max-width: 600px) 100vw, 600px\" \/><\/figure><\/div>\n\n\n\n<p>Esse \u00e9 o mais diferente de todos pois n\u00e3o possui faces com planos definidos!<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>At = 4 . \u03c0 . r\u00b2<\/strong><\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>V = 4 \u2044 3. \u03c0 . r\u00b3<\/strong><\/p>\n\n\n\n<p>Gostou do nosso artigo sobre Geometria Espacial? Confira <strong>outros artigos <\/strong>do nosso blog e <strong>se prepare para o Enem<\/strong> da melhor maneira! Voc\u00ea tamb\u00e9m pode se organizar com o nosso<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>, o mais completo da internet, e o melhor: <strong>totalmente gratuito!<\/strong><\/p>\n\n\n\n<p>Queremos te ajudar a encontrar a FACULDADE IDEAL! Logo abaixo, fa\u00e7a uma pesquisa por curso e cidade que te mostraremos todas as faculdades que podem te atender. Informamos a nota de corte, valor de mensalidade, nota do MEC, avalia\u00e7\u00e3o dos alunos, modalidades de ensino e muito mais.<\/p>\n\n\n\n<p>Experimente agora!<\/p>\n","protected":false},"excerpt":{"rendered":"<p>A Geometria Espacial \u00e9 a \u00e1rea que estuda os s\u00f3lidos geom\u00e9tricos, ou seja, aqueles que possuem tr\u00eas dimens\u00f5es e n\u00e3o &#8220;cabem&#8221; no papel. Al\u00e9m de calcular a \u00e1rea, aprendemos uma novidade: como encontrar o volume. Neste resumo, voc\u00ea aprende tudo o que \u00e9 importante saber: defini\u00e7\u00f5es, exemplos e f\u00f3rmulas. Neste artigo sobre Geometria Espacial voc\u00ea [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":17382,"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":[487],"tags":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v20.10 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Aprenda Geometria Espacial: defini\u00e7\u00f5es, exemplos e f\u00f3rmulas!<\/title>\n<meta name=\"description\" content=\"A Geometria Espacial estuda as figuras 3D chamadas de s\u00f3lidos. Descubra quais s\u00e3o eles e seus tipos. 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