pirâmides e prisma
Autor: B.C.D.D.J. | Publicado: 2011-11-29T14:40:00-08:00

Prismas  &nbsp;  Ø&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Um prisma é um poliedro limitado por dois polígonos e paralelos (as bases) e vários paralelogramos (as faces laterais).  Ø&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; A altura do prisma é a distância entre as bases.  Ø&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Se todas as faces laterais são rectângulos, elas serão perpendiculares às bases e então o prisma chama-se prisma recto.  Ø&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Se as faces laterais não são perpendiculares às bases, chama-se prisma oblíquo.  Ø&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Um prisma é regular quando tem um prisma recto que cujas bases são polígonos regulares.  Ø&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; As&nbsp; arestas laterais de um prisma são segmentos iguais e paralelos entre si. Nos prismas rectos são prependiculares às bases.  &nbsp;  
  
     
 
 
 
 
 
  

&nbsp;  Classificação dos prismas segundo o polígono das bases&nbsp;  Conforme os polígonos das bases são triângulos, quadriláteros, pentágonos, etc, o prisma chama-se triangular, quadrangular, pentagonal, etc.  Os prismas rectos cujas bases são polígonos regulares chamam-se prismas regulares.  Quer em objectos de uso corrente, quer na Natureza, encontramos com frequência formas prismáticas.&nbsp;  &nbsp;  
Área do prisma&nbsp;  

            Área lateral  =&nbsp;       Perímetro da base x altura              
Área total  =       Área lateral +2 x Área da base               
   
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Volume do prisma&nbsp;  
O volume de um prisma recto ou oblíquo, é:  &nbsp;  
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            Volume       do prisma =       Área da base x altura              
   
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Recordamos que a&nbsp; altura de um prisma é a distância entre as duas bases. Se o prisma é recto, a altura coincide com o comprimento das arestas laterais.  
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Planificação de prismas&nbsp;    


Pirâmides&nbsp;  Ø&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Uma pirâmide é um poliedro que tem por base um polígono qualquer e por faces laterais triângulos com um vértice comum, que se&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; chama vértice da piramide.  Ø&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; A altura da pirâmide é a distância do vértice ao plano da base.  Ø&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Uma pirâmide é regular quando a base é um polígono regular e o vértice projecta-se sobre o centro desse polígono.  Ø&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Uma pirâmide é oblíqua quando a pojecção do vértice não coincide com o cento do polígono da base.  Ø&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Uma pirâmide é recta quando o vértice tem a sua projecção coincidente com o centro da base.  Ø&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Numa pirâmide regular as arestas laterais são todas iguais e as faces são triângulos isósceles iguais. As alturas desses triângulos&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; chamam-se apótemas da&nbsp;pirâmide.  Ø&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; O apótema de uma pirâmide regular é a hipotenusa de um triângulo rectangulo cujos catetos são a altura da pirâmide e o apótema do&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; polígono da base.  Ø&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; As pirâmides chamam-se triangulares, quadrangulares, pentagonais,... consoante o polígono da base seja um triângulo, um&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; quadrilátero, um pentagono,...   &nbsp;  Podemos observar formas piramidais em obejectos reais, em constuções...  As pirâmides do Egipto construidas muitos séculos antes da nossa era como sepulcro dos faraós, são quadrangula e as bases estão orientadas segundo os&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; pontos cardeais.  &nbsp;  &nbsp;  &nbsp;  Área da pirâmide&nbsp;  A área lateral de uma pirâmide é a soma de n triâgulos iguais :  &nbsp;  &nbsp;  
            Área lateral       = n x 1 &nbsp;la       = 1 (nl) x a = Perímetro da base x a&nbsp;&nbsp;&nbsp;       &nbsp;              &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;       &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;2&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;       2&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;       2                 
&nbsp;  &nbsp;  Como a base é um polígono regular, a sua área é&nbsp;&nbsp;&nbsp;&nbsp; Perímetro da base x a’  &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;  2   onde a’ é o apótema da base.   
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            Área       total=área lateral+área da base=Perímetro base x a+Perímetro       base x       a'       &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;       2&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;       &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;       &nbsp;&nbsp;&nbsp; 2                 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;   &nbsp;  &nbsp;  &nbsp;Volume da pirâmide     &nbsp;  Temos um prisma e uma pirâmide com a mesma base e a mesma altura.   Vamos comparar os seus volumes.  &nbsp;    &nbsp;  &nbsp;  Se enchermos de água e vertermos dentro do prisma, ficará cheia uma terça parte deste. Quer dizer, são necessárias três pirâmides para completar o volume do prisma       &nbsp;  
            Volume       da pirâmide =&nbsp; 1       Área da base x altura              &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;              3                 
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Planificação de pirâmides    &nbsp;    
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