[justify]Considere a pirâmide P, de altura h e base [math]ABCDE[/math] de área [math]A1[/math], contida em um plano horizontal [math]\alpha[/math], e um plano [math]\beta[/math], paralelo a [math]\alpha[/math]e secante à pirâmide. O plano [math]\beta[/math] determina uma secção transversal [math]abcde[/math] de área [math]A2[/math], que é base da pirâmide Q de altura d (pirâmide menor) e semelhante à base [math]ABCDE[/math]. Se os lado [math]AB[/math] da pirâmide P e os lados [math]ab[/math] da pirâmide Q são os comprimentos dos lados correspondentes de dois polígonos semelhantes das áreas[math]A1[/math] e [math]A2[/math]´, então:[br][br][br][math]\frac{A1}{A2}=\left(\frac{AB}{ab}\right)^2[/math][/justify]

[justify]Note, na figura anterior, que as respectivas faces laterais das pirâmides Q e P são triângulos semelhantes. Usando semelhança de triângulos, pode-se demonstrar que as bases de áreas [math]A1[/math] e [math]A2[/math] são polígonos semelhantes, com razão de semelhança [br][/justify][math]\frac{h}{d}[/math] , então, [math]\frac{A1}{A2}=\left(\frac{h}{d}\right)^2[/math][br][br]

[justify]Agora, considere duas pirâmides M e N de mesma medida h de altura, com bases de mesma área [math]A_M[/math] e [math]A_N[/math] contidas em um plano horizontal [math]\alpha[/math]. Qualquer plano [math]\beta[/math], paralelo a [math]\alpha[/math] e secante às pirâmides, determina duas secções transversais de áreas [math]S_M[/math] e [math]S_N[/math], respectivamente. Veja o exemplo na janela 3d abaixo.[/justify]

Vimos anteriormente que a razão entre a área da base e da secção transversal de cada pirâmide vale[br][br][math]\frac{A_M}{A_M}=\left(\frac{h}{d}\right)^2[/math] e [math]\frac{A_N}{S_N}=\left(\frac{h}{d}\right)^2[/math].[br][br]Logo, = [math]\frac{A_M}{S_M}=\frac{A_N}{S_N}[/math].[br][justify][br]Como [math]A_M[/math] = [math]A_N[/math], concluímos que [math]S_M[/math]= [math]S_N[/math] para qualquer plano [math]\beta[/math] paralelo a [math]\alpha[/math]. Com isso, pelo princípio de Cavalieri, como todas as secções transversais de duas pirâmides de mesma altura têm áreas iguais, então seus volumes são iguais. Provamos, assim, que duas pirâmides de mesma base e com mesma altura têm volumes iguais. Esse fato será utilizado a seguir para determinar o volume de uma pirâmide.[/justify]

Para calcular o volume de uma pirâmide qualquer, primeiramente, vamos considerar o prisma reto de base triangular. Esse prisma pode ser decomposto em três pirâmides triangulares, como mostram [br]as figuras a seguir. a seguir.

[b][size=150]Observe que:[/size][/b][br][br][justify]• as pirâmides [b]I[/b] e [b]II [/b]têm a mesma altura (altura do prisma), têm bases congruentes [math](\Delta ABC\cong\Delta DEF[/math], pois cada triângulo é uma base do prisma) e, portanto, as pirâmides[b] I[/b] e [b]II [/b]têm o mesmo volume;[br][br]• considerando as pirâmides [b]II[/b] e [b]III[/b] com respectivas bases [math]CEF[/math] e [math]BCE[/math], a altura (distância do ponto [math]D[/math]ao retângulo [math]BCFE[/math]) dessas pirâmides é a mesma, têm bases congruentes ([math]\Delta CEF\cong\Delta BCE[/math], pois cada um desses triângulos é a metade do retângulo [math]BCFE[/math]) e, portanto, novamente, as pirâmides [b]II[/b] e [b]III [/b]têm o mesmo volume. [br][br]Logo, as pirâmides [b]I[/b],[b] II[/b] e[b] III [/b]têm o mesmo volume, ou seja, [math]V_1=V_2=V_3[/math].[/justify]Seja [math]V_{prisma}[/math]= [math]V_1+V_2+V_3[/math](soma dos volumes das três pirâmides) e considerando [math]V_1=V_2=V_3=V[/math], temos:[br][br][br][math]V_{prisma}=V+V+V\Longrightarrow V_{prisma}=3V\Longrightarrow V=\frac{V_{prisma}}{3}[/math][br][br]Assim, o volume de cada pirâmide é igual a [math]\frac{1}{3}[/math] do volume do prisma triangular dado. Como o volume do prisma é [math]V_{prisma}=A_b.h[/math], podemos escrever: [math]V=\frac{V_{prisma}}{3}[/math] ou [math]\Longrightarrow V=\frac{A_b.h}{3}[/math][br][br]

[justify][img]data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJAAAAC6CAYAAABBccToAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsMAAA7DAcdvqGQAAB34aVRYdFNuaXBNZXRhZGF0YQAAAAAAeyJjbGlwUG9pbnRzIjpbeyJ4Ijo0MywieSI6MH0seyJ4Ijo0NCwieSI6MH0seyJ4Ijo0NCwieSI6MX0seyJ4Ijo0NSwieSI6MX0seyJ4Ijo0NywieSI6Mn0seyJ4Ijo0OCwieSI6Mn0seyJ4Ijo0OSwieSI6M30seyJ4Ijo1MCwieSI6M30seyJ4Ijo1MSwieSI6M30seyJ4Ijo1MiwieSI6M30seyJ4Ijo1MywieSI6NH0seyJ4Ijo1NCwieSI6NH0seyJ4Ijo1NiwieSI6NX0seyJ4Ijo1NywieSI6Nn0seyJ4Ijo1OSwieSI6Nn0seyJ4Ijo2MCwieSI6N30seyJ4Ijo2MiwieSI6OH0seyJ4Ijo2MywieSI6OH0seyJ4Ijo2NSwieSI6OX0seyJ4Ijo2NiwieSI6MTB9LHsieCI6NjcsInkiOjEwfSx7IngiOjY4LCJ5IjoxMX0seyJ4Ijo2OSwieSI6MTF9LHsieCI6NzAsInkiOjExfSx7IngiOjcxLCJ5IjoxMX0seyJ4Ijo3MiwieSI6MTF9LHsieCI6NzMsInkiOjExfSx7IngiOjc0LCJ5IjoxMX0seyJ4Ijo3NSwieSI6MTF9LHsieCI6NzYsInkiOjExfSx7IngiOjc3LCJ5IjoxMX0seyJ4Ijo3OCwieSI6MTF9LHsieCI6NzksInkiOjExfSx7IngiOjgwLCJ5IjoxMX0seyJ4Ijo4MSwieSI6MTF9LHsieCI6ODIsInkiOjEyfSx7IngiOjgzLCJ5IjoxMn0seyJ4Ijo4NSwieSI6MTN9LHsieCI6ODYsInkiOjEzfSx7IngiOjg3LCJ5IjoxNH0seyJ4Ijo4OSwieSI6MTV9LHsieCI6OTAsInkiOjE2fSx7IngiOjkyLCJ5IjoxN30seyJ4Ijo5MywieSI6MTh9LHsieCI6OTQsInkiOjIwfSx7IngiOjk2LCJ5IjoyMX0seyJ4Ijo5NywieSI6MjN9LHsieCI6OTgsInkiOjI1fSx7IngiOjEwMCwieSI6Mjd9LHsieCI6MTAyLCJ5IjozMX0seyJ4IjoxMDQsInkiOjM0fSx7IngiOjEwNiwieSI6Mzd9LHsieCI6MTA4LCJ5Ijo0MH0seyJ4IjoxMDksInkiOjQzfSx7IngiOjExMCwieSI6NDV9LHsieCI6MTExLCJ5Ijo0Nn0seyJ4IjoxMTIsInkiOjQ4fSx7IngiOjExMywieSI6NDl9LHsieCI6MTE0LCJ5Ijo1MH0seyJ4IjoxMTQsInkiOjUxfSx7IngiOjExNSwieSI6NTJ9LHsieCI6MTE2LCJ5Ijo1M30seyJ4IjoxMTYsInkiOjU0fSx7IngiOjExNywieSI6NTV9LHsieCI6MTE4LCJ5Ijo1Nn0seyJ4IjoxMTksInkiOjU3fSx7IngiOjEyMCwieSI6NTh9LHsieCI6MTIxLCJ5Ijo2MH0seyJ4IjoxMjIsInkiOjYxfSx7IngiOjEyMywieSI6NjJ9LHsieCI6MTIzLCJ5Ijo2M30seyJ4IjoxMjQsInkiOjY0fSx7IngiOjEyNSwieSI6NjV9LHsieCI6MTI1LCJ5Ijo2Nn0seyJ4IjoxMjYsInkiOjY2fSx7IngiOjEyNiwieSI6Njd9LHsieCI6MTI3LCJ5Ijo2OH0seyJ4IjoxMjcsInkiOjY5fSx7IngiOjEyOCwieSI6NzB9LHsieCI6MTI4LCJ5Ijo3MX0seyJ4IjoxMjksInkiOjcyfSx7IngiOjEyOSwieSI6NzR9LHsieCI6MTI5LCJ5Ijo3Nn0seyJ4IjoxMzAsInkiOjc3fSx7IngiOjEzMCwieSI6ODB9LHsieCI6MTMwLCJ5Ijo4Mn0seyJ4IjoxMzEsInkiOjg0fSx7IngiOjEzMSwieSI6ODd9LHsieCI6MTMxLCJ5Ijo5MH0seyJ4IjoxMzEsInkiOjkyfSx7IngiOjEzMSwieSI6OTR9LHsieCI6MTMxLCJ5Ijo5N30seyJ4IjoxMzEsInkiOjk5fSx7IngiOjEzMSwieSI6MTAyfSx7IngiOjEzMSwieSI6MTA0fSx7IngiOjEzMSwieSI6MTA2fSx7IngiOjEzMSwieSI6MTA4fSx7IngiOjEzMSwieSI6MTA5fSx7IngiOjEzMSwieSI6MTExfSx7IngiOjEzMSwieSI6MTEzfSx7IngiOjEzMSwieSI6MTE1fSx7IngiOjEzMSwieSI6MTE3fSx7IngiOjEzMSwieSI6MTE5fSx7IngiOjEzMSwieSI6MTIwfSx7IngiOjEzMSwieSI6MTIyfSx7IngiOjEzMSwieSI6MTIzfSx7IngiOjEzMiwieSI6MTI0fSx7IngiOjEzMiwieSI6MTI1fSx7IngiOjEzMywieSI6MTI2fSx7IngiOjEzNCwieSI6MTI4fSx7IngiOjEzNSwieSI6MTI5fSx7IngiOjEzNiwieSI6MTMxfSx7IngiOjEzNywieSI6MTMyfSx7IngiOjEzOCwieSI6MTMzfSx7IngiOjEzOCwieSI6MTM0fSx7IngiOjEzOSwieSI6MTM1fSx7IngiOjE0MCwieSI6MTM2fSx7IngiOjE0MSwieSI6MTM3fSx7IngiOjE0MiwieSI6MTM4fSx7IngiOjE0MywieSI6MTM5fSx7IngiOjE0MywieSI6MTQwfSx7IngiOjE0NCwieSI6MTQyfSx7IngiOjE0NCwieSI6MTQzfSx7IngiOjE0NCwieSI6MTQ0fSx7IngiOjE0NCwieSI6MTQ1fSx7IngiOjE0NCwieSI6MTQ2fSx7IngiOjE0NCwieSI6MTQ3fSx7IngiOjE0NCwieSI6MTQ4fSx7IngiOjE0NCwieSI6MTQ5fSx7IngiOjE0NCwieSI6MTUwfSx7IngiOjE0MywieSI6MTUxfSx7IngiOjE0MywieSI6MTUyfSx7IngiOjE0MiwieSI6MTU0fSx7IngiOjE0MSwieSI6MTU1fSx7IngiOjEzOSwieSI6MTU2fSx7IngiOjEzOCwieSI6MTU2fSx7IngiOjEzNywieSI6MTU3fSx7IngiOjEzNiwieSI6MTU4fSx7IngiOjEzNSwieSI6MTU4fSx7IngiOjEzMywieSI6MTU5fSx7IngiOjEzMiwieSI6MTYwfSx7IngiOjEzMSwieSI6MTYwfSx7IngiOjEyOSwieSI6MTYxfSx7IngiOjEyOCwieSI6MTYxfSx7IngiOjEyNywieSI6MTYyfSx7IngiOjEyNSwieSI6MTYyfSx7IngiOjEyNCwieSI6MTYzfSx7IngiOjEyMiwieSI6MTYzfSx7IngiOjEyMCwieSI6MTY0fSx7IngiOjExOSwieSI6MTY0fSx7IngiOjExNywieSI6MTY1fSx7IngiOjExNiwieSI6MTY1fSx7IngiOjExNSwieSI6MTY1fSx7IngiOjExNCwieSI6MTY2fSx7IngiOjExMywieSI6MTY2fSx7IngiOjExMiwieSI6MTY2fSx7IngiOjExMSwieSI6MTY2fSx7IngiOjExMCwieSI6MTY2fSx7IngiOjEwOSwieSI6MTY2fSx7IngiOjEwOCwieSI6MTY2fSx7IngiOjEwNywieSI6MTY3fSx7IngiOjEwNiwieSI6MTY3fSx7IngiOjEwNSwieSI6MTY3fSx7IngiOjEwNCwieSI6MTY3fSx7IngiOjEwMywieSI6MTY3fSx7IngiOjEwMiwieSI6MTY3fSx7IngiOjEwMSwieSI6MTY3fSx7IngiOjEwMCwieSI6MTY3fSx7IngiOjk5LCJ5IjoxNjd9LHsieCI6OTgsInkiOjE2N30seyJ4Ijo5NywieSI6MTY3fSx7IngiOjk3LCJ5IjoxNjh9LHsieCI6OTYsInkiOjE2OH0seyJ4Ijo5NiwieSI6MTY5fSx7IngiOjk1LCJ5IjoxNzB9LHsieCI6OTUsInkiOjE3Mn0seyJ4Ijo5NSwieSI6MTczfSx7IngiOjk0LCJ5IjoxNzR9LHsieCI6OTQsInkiOjE3NX0seyJ4Ijo5NCwieSI6MTc2fSx7IngiOjkzLCJ5IjoxNzd9LHsieCI6OTMsInkiOjE3OH0seyJ4Ijo5MywieSI6MTc5fSx7IngiOjkzLCJ5IjoxODB9LHsieCI6OTIsInkiOjE4MX0seyJ4Ijo5MiwieSI6MTgyfSx7IngiOjkxLCJ5IjoxODN9LHsieCI6OTEsInkiOjE4NH0seyJ4Ijo5MCwieSI6MTg0fSx7IngiOjg5LCJ5IjoxODR9LHsieCI6ODgsInkiOjE4NH0seyJ4Ijo4NywieSI6MTg0fSx7IngiOjg2LCJ5IjoxODV9LHsieCI6ODUsInkiOjE4NX0seyJ4Ijo4NCwieSI6MTg1fSx7IngiOjgzLCJ5IjoxODZ9LHsieCI6ODIsInkiOjE4Nn0seyJ4Ijo4MSwieSI6MTg2fSx7IngiOjgwLCJ5IjoxODZ9LHsieCI6NzksInkiOjE4Nn0seyJ4Ijo3OCwieSI6MTg2fSx7IngiOjc3LCJ5IjoxODZ9LHsieCI6NzYsInkiOjE4Nn0seyJ4Ijo3NSwieSI6MTg2fSx7IngiOjc0LCJ5IjoxODZ9LHsieCI6NzMsInkiOjE4NX0seyJ4Ijo3MiwieSI6MTg1fSx7IngiOjcyLCJ5IjoxODR9LHsieCI6NzEsInkiOjE4NH0seyJ4Ijo3MCwieSI6MTgzfSx7IngiOjY5LCJ5IjoxODF9LHsieCI6NjgsInkiOjE4MH0seyJ4Ijo2NywieSI6MTc5fSx7IngiOjY3LCJ5IjoxNzh9LHsieCI6NjYsInkiOjE3N30seyJ4Ijo2NSwieSI6MTc2fSx7IngiOjY1LCJ5IjoxNzV9LHsieCI6NjQsInkiOjE3NH0seyJ4Ijo2MywieSI6MTczfSx7IngiOjYyLCJ5IjoxNzN9LHsieCI6NjEsInkiOjE3M30seyJ4Ijo2MCwieSI6MTcyfSx7IngiOjU5LCJ5IjoxNzJ9LHsieCI6NTcsInkiOjE3MX0seyJ4Ijo1NSwieSI6MTcxfSx7IngiOjU0LCJ5IjoxNzF9LHsieCI6NTIsInkiOjE3MH0seyJ4Ijo1MCwieSI6MTcwfSx7IngiOjQ5LCJ5IjoxNzB9LHsieCI6NDgsInkiOjE3MH0seyJ4Ijo0NywieSI6MTcwfSx7IngiOjQ2LCJ5IjoxNzB9LHsieCI6NDYsInkiOjE3MX0seyJ4Ijo0NiwieSI6MTcyfSx7IngiOjQ2LCJ5IjoxNzN9LHsieCI6NDYsInkiOjE3NH0seyJ4Ijo0NiwieSI6MTc1fSx7IngiOjQ1LCJ5IjoxNzZ9LHsieCI6NDUsInkiOjE3N30seyJ4Ijo0NCwieSI6MTc4fSx7IngiOjQzLCJ5IjoxNzh9LHsieCI6NDIsInkiOjE3OX0seyJ4Ijo0MSwieSI6MTgwfSx7IngiOjQwLCJ5IjoxODF9LHsieCI6MzgsInkiOjE4MX0seyJ4IjozNywieSI6MTgyfSx7IngiOjM2LCJ5IjoxODJ9LHsieCI6MzUsInkiOjE4M30seyJ4IjozMywieSI6MTg0fSx7IngiOjMyLCJ5IjoxODR9LHsieCI6MzEsInkiOjE4NX0seyJ4IjozMCwieSI6MTg1fSx7IngiOjI5LCJ5IjoxODV9LHsieCI6MjcsInkiOjE4NX0seyJ4IjoyNiwieSI6MTg0fSx7IngiOjI1LCJ5IjoxODN9LHsieCI6MjQsInkiOjE4Mn0seyJ4IjoyNCwieSI6MTgwfSx7IngiOjI0LCJ5IjoxNzl9LHsieCI6MjQsInkiOjE3OH0seyJ4IjoyNCwieSI6MTc2fSx7IngiOjI0LCJ5IjoxNzV9LHsieCI6MjQsInkiOjE3NH0seyJ4IjoyNCwieSI6MTczfSx7IngiOjI0LCJ5IjoxNzJ9LHsieCI6MjQsInkiOjE3MX0seyJ4IjoyNCwieSI6MTcwfSx7IngiOjI0LCJ5IjoxNjl9LHsieCI6MjQsInkiOjE2OH0seyJ4IjoyNCwieSI6MTY3fSx7IngiOjIzLCJ5IjoxNjV9LHsieCI6MjMsInkiOjE2NH0seyJ4IjoyMiwieSI6MTYzfSx7IngiOjIyLCJ5IjoxNjF9LHsieCI6MjEsInkiOjE2MH0seyJ4IjoxOSwieSI6MTYwfSx7IngiOjE4LCJ5IjoxNTl9LHsieCI6MTcsInkiOjE1OX0seyJ4IjoxNiwieSI6MTU5fSx7IngiOjE1LCJ5IjoxNTl9LHsieCI6MTQsInkiOjE1OX0seyJ4IjoxMiwieSI6MTU5fSx7IngiOjExLCJ5IjoxNTl9LHsieCI6MTAsInkiOjE1OX0seyJ4Ijo5LCJ5IjoxNTl9LHsieCI6OCwieSI6MTU4fSx7IngiOjYsInkiOjE1OH0seyJ4Ijo1LCJ5IjoxNTd9LHsieCI6NCwieSI6MTU2fSx7IngiOjMsInkiOjE1Nn0seyJ4IjoyLCJ5IjoxNTV9LHsieCI6MSwieSI6MTU0fSx7IngiOjAsInkiOjE1M30seyJ4IjowLCJ5IjoxNTJ9LHsieCI6MCwieSI6MTUxfSx7IngiOjAsInkiOjE1MH0seyJ4IjowLCJ5IjoxNDl9LHsieCI6MSwieSI6MTQ4fSx7IngiOjIsInkiOjE0N30seyJ4IjoyLCJ5IjoxNDZ9LHsieCI6MywieSI6MTQ1fSx7IngiOjMsInkiOjE0NH0seyJ4Ijo0LCJ5IjoxNDJ9LHsieCI6NSwieSI6MTQxfSx7IngiOjYsInkiOjE0MH0seyJ4Ijo2LCJ5IjoxMzl9LHsieCI6NywieSI6MTM4fSx7IngiOjgsInkiOjEzN30seyJ4Ijo4LCJ5IjoxMzZ9LHsieCI6OSwieSI6MTM1fSx7IngiOjEwLCJ5IjoxMzR9LHsieCI6MTAsInkiOjEzM30seyJ4IjoxMSwieSI6MTMyfSx7IngiOjExLCJ5IjoxMzF9LHsieCI6MTIsInkiOjEzMH0seyJ4IjoxMiwieSI6MTI5fSx7IngiOjEzLCJ5IjoxMjh9LHsieCI6MTMsInkiOjEyN30seyJ4IjoxNCwieSI6MTI2fSx7IngiOjE1LCJ5IjoxMjV9LHsieCI6MTUsInkiOjEyM30seyJ4IjoxNiwieSI6MTIyfSx7IngiOjE3LCJ5IjoxMjF9LHsieCI6MTgsInkiOjExOX0seyJ4IjoxOSwieSI6MTE4fSx7IngiOjIwLCJ5IjoxMTd9LHsieCI6MjAsInkiOjExNn0seyJ4IjoyMSwieSI6MTE1fSx7IngiOjIxLCJ5IjoxMTR9LHsieCI6MjIsInkiOjExM30seyJ4IjoyMiwieSI6MTEyfSx7IngiOjIzLCJ5IjoxMTJ9LHsieCI6MjMsInkiOjExMX0seyJ4IjoyMywieSI6MTEwfSx7IngiOjI0LCJ5IjoxMDl9LHsieCI6MjQsInkiOjEwOH0seyJ4IjoyNSwieSI6MTA3fSx7IngiOjI1LCJ5IjoxMDZ9LHsieCI6MjYsInkiOjEwNX0seyJ4IjoyNiwieSI6MTA0fSx7IngiOjI3LCJ5IjoxMDJ9LHsieCI6MjgsInkiOjEwMX0seyJ4IjoyOCwieSI6MTAwfSx7IngiOjI5LCJ5Ijo5OX0seyJ4IjoyOSwieSI6OTh9LHsieCI6MzAsInkiOjk2fSx7IngiOjMwLCJ5Ijo5NX0seyJ4IjozMSwieSI6OTR9LHsieCI6MzEsInkiOjkyfSx7IngiOjMyLCJ5Ijo5MX0seyJ4IjozMywieSI6OTB9LHsieCI6MzMsInkiOjg5fSx7IngiOjM0LCJ5Ijo4OH0seyJ4IjozNCwieSI6ODd9LHsieCI6MzQsInkiOjg2fSx7IngiOjM0LCJ5Ijo4NX0seyJ4IjozNCwieSI6ODR9LHsieCI6MzQsInkiOjgzfSx7IngiOjM0LCJ5Ijo4Mn0seyJ4IjozNSwieSI6ODF9LHsieCI6MzUsInkiOjgwfSx7IngiOjM1LCJ5Ijo3OX0seyJ4IjozNiwieSI6Nzh9LHsieCI6MzYsInkiOjc3fSx7IngiOjM2LCJ5Ijo3Nn0seyJ4IjozNywieSI6NzV9LHsieCI6MzcsInkiOjczfSx7IngiOjM3LCJ5Ijo3Mn0seyJ4IjozNywieSI6NzF9LHsieCI6MzcsInkiOjcwfSx7IngiOjM3LCJ5Ijo2OX0seyJ4IjozNywieSI6Njh9LHsieCI6MzcsInkiOjY3fSx7IngiOjM4LCJ5Ijo2Nn0seyJ4IjozOCwieSI6NjV9LHsieCI6MzgsInkiOjY0fSx7IngiOjM4LCJ5Ijo2M30seyJ4IjozOCwieSI6NjJ9LHsieCI6MzksInkiOjYxfSx7IngiOjM5LCJ5Ijo2MH0seyJ4IjozOSwieSI6NTl9LHsieCI6NDAsInkiOjU3fSx7IngiOjQwLCJ5Ijo1Nn0seyJ4Ijo0MCwieSI6NTR9LHsieCI6NDEsInkiOjUyfSx7IngiOjQxLCJ5Ijo1MX0seyJ4Ijo0MSwieSI6NDl9LHsieCI6NDIsInkiOjQ4fSx7IngiOjQyLCJ5Ijo0N30seyJ4Ijo0MiwieSI6NDZ9LHsieCI6NDIsInkiOjQ1fSx7IngiOjQzLCJ5Ijo0NH0seyJ4Ijo0MywieSI6NDN9LHsieCI6NDMsInkiOjQyfSx7IngiOjQ0LCJ5Ijo0Mn0seyJ4Ijo0NCwieSI6NDF9LHsieCI6NDQsInkiOjM5fSx7IngiOjQ0LCJ5IjozOH0seyJ4Ijo0NCwieSI6MzZ9LHsieCI6NDQsInkiOjM1fSx7IngiOjQ0LCJ5IjozNH0seyJ4Ijo0NCwieSI6MzN9LHsieCI6NDUsInkiOjMyfSx7IngiOjQ1LCJ5IjozMX0seyJ4Ijo0NSwieSI6MzB9LHsieCI6NDUsInkiOjI5fSx7IngiOjQ1LCJ5IjoyOH0seyJ4Ijo0NSwieSI6Mjd9LHsieCI6NDYsInkiOjI2fSx7IngiOjQ2LCJ5IjoyNX0seyJ4Ijo0NiwieSI6MjR9LHsieCI6NDYsInkiOjIzfSx7IngiOjQ2LCJ5IjoyMn0seyJ4Ijo0NiwieSI6MjF9LHsieCI6NDYsInkiOjIwfSx7IngiOjQ2LCJ5IjoxOX0seyJ4Ijo0NywieSI6MTl9LHsieCI6NDcsInkiOjE4fSx7IngiOjQ4LCJ5IjoxOH0seyJ4Ijo0OSwieSI6MTh9LHsieCI6NTAsInkiOjE4fSx7IngiOjUxLCJ5IjoxOH0seyJ4Ijo1MiwieSI6MTd9LHsieCI6NTMsInkiOjE3fSx7IngiOjU0LCJ5IjoxN30seyJ4Ijo1NSwieSI6MTd9LHsieCI6NTYsInkiOjE3fSx7IngiOjU3LCJ5IjoxNn0seyJ4Ijo1OCwieSI6MTZ9LHsieCI6NTksInkiOjE1fSx7IngiOjYwLCJ5IjoxNH0seyJ4Ijo2MSwieSI6MTN9LHsieCI6NjEsInkiOjEyfSx7IngiOjYyLCJ5IjoxMX0seyJ4Ijo2MiwieSI6MTB9LHsieCI6NjIsInkiOjl9LHsieCI6NjMsInkiOjh9LHsieCI6NjMsInkiOjd9LHsieCI6NjMsInkiOjZ9LHsieCI6NjQsInkiOjZ9LHsieCI6NjMsInkiOjV9LHsieCI6NjIsInkiOjZ9LHsieCI6NjAsInkiOjZ9LHsieCI6NTgsInkiOjd9LHsieCI6NTcsInkiOjh9LHsieCI6NTUsInkiOjl9LHsieCI6NTQsInkiOjl9LHsieCI6NTMsInkiOjEwfSx7IngiOjUzLCJ5IjoxMX0seyJ4Ijo1NCwieSI6MTF9LHsieCI6NTUsInkiOjExfSx7IngiOjU3LCJ5IjoxMX0seyJ4Ijo1OCwieSI6MTF9LHsieCI6NTksInkiOjExfSx7IngiOjYwLCJ5IjoxMX0seyJ4Ijo2MSwieSI6MTB9LHsieCI6NjEsInkiOjl9LHsieCI6NjIsInkiOjl9XX0ZfaI3AAAe2UlEQVR4Xu2dfVxUVf7HP0NmIaiZMqDSrqFBgqgJlrpm5ehaFrpbYWlZWaSttWnWPvzU9SHL0gR8+GWJDFiW5oxa6pYpor8tS0gtSwcTV6yEAUUeHBhAoXt+f8zc6c5hhrn3zn0anPfrxR+cc3gaPvM9n3O+95wvEEAQQu5mGOYBuj2IeujohkCAEHIDgA4AanU6XQvdH0Q5AlJALAzD9HMKqEKn0zXR/UGC8IIQEsUwTCbDMI0Mw8ym+4ME4QUh5AaGYRYyDFPDMMwBQsjd9JggQXxCCAlnGGY2wzDl3iKSM2J57AsSBHAI6XqGYdIYhjnNMMwBhmFmEw70+CBBPMIwzFsMw2wmhAx2bgUcCAooiE+4YmFhGOZjZ3tm0CcF8QjDMAu5oqGBQ1wuGIZZSH+PIFcxXHHQsMaZbiecviBB2sRXhBLK1RzBAnonWixEPgNtBmDW6XRmuiNIO0Hq6OONqzkqtVvEiqeiooJcvnyZbiYV5eUe2zmYCCGp9O8RJAARKx5CCBk+9Hby2aefurUd2L+fDIxPILW1tW7tnmjP0SiEbmhvsMLR6XSL6D6+ROj1qLpY5db2zttr8eS0aejatatbuyd0Ot2i9hqN2pWAPEUZf4TDotfrUVX1m4AOf/MNThYV4em0Z9zG+SAVgKm9RaOAFpAcYvGEQ0AXXZ+vffttPPHUk7jhhhvcxvHBGY3aTSQKWAER+ZbireBOYUUWC44cPoJn0tLoYUIICuhqQq/Xo7raIaB33l6Lx6dORbcbb6SHCSG1vXqigIGevuQkb89eMvae0eSns2dJYv94UnWxih7iDwEtomAE4kGEXo/a2lpsyN2AKY89hhu7+xV9aAJaQAEL/TaWE2uZlSTE3UqGD72dVFZW0t1SELAiCkYgHkToI9DU1IT7Ux5Ajx496G4pCFgBBWwylSi4ClMCQsiikJCQxXS71gnICNTeNuPg3B8KxL8rICNQe4s+FJMC6XGQgIxAavDDd9/STXIRsH4oYKCXMHLz2b93kTuGDCFr16ymu+QiYEQUnMJ4MHhAIurq6gAA//nqIKKjo+khUmPW6XST6EYtEpzCfLDNvBUMGNfnZtMWt36ZSCWETKcbtUjACUjplYrRmA17nR0A0KHDNcg15sBms9HD5GA6ISSJbtQaAScguR7Z8ERhQQFOnfwRAKDvGYGbYnqDIQxyjTn0UDlICoQoFHACUpLMjAzAKZ7InhG4tmMHGCb8AaYtm+mhsqDT6aZrXURBAXmhqKgIhwu/QeeunTEwKQEA0NzcjBGGZFSUn8c281b6S+RC01NZUEBeyMnORr+4W1B3qQ6JTgHZaurRXd8Nw0cnITMjnf4SudD0VBYUkAdKS0vx8bbtuLGH47GNgUkJiOylR0vzrwCAlCljUW4tR96evdRXyoOWp7KggDywKiMTUb2icOXyFURE9UBkrwhE9ooAwziW89313TDojnhkrX+X/lI50eRUFhQQhc1mw969ezBtZhp++elnDEoeQA8BAIyZcCe+PfwtCgsK6C650ORUFhQQRa4xByDAnaPvQvXFapf/AYCQkN9ertjEGMQOiEH2+ixXm9w4pzJN3WcUFBAHm82GHKMRqVMfwe5PPgWc/oflmg7XcEYDww1J2L9vP0pLS93a5YQQchfdpiYBJSC5c2B5e/aivq4OqVMfxZf7/+PyPyyN9ka38SMMyeiu74aVzv0ihQhGIK2yMjMD904cj/DO4a38T2JSAhobWt9l/sDkMfh423al0hvQ6XR3aylbHzACkjsHts28FdYyK56e6TgwSPsflnMlVrfPRxiSERp2PYzZ693aZSYoIKHInQMzm024d+J4RPXqCfNGR8ad639YGu2to5Bhwkglk6wAkCr3G4ovASMgOSksKMDhwm9w38T7AcCj/2kLw4SRYAijZHpDM2fsgwICsDIjE4OTb8NtQ4cAQCv/w+XUiTN0EzqFhWLE6CRkr19Hd8lNUEB8kDNcl5aW4pvCQkxzeh948T99Y/sAAErPlru1sxgmjlQ6yQotTGUBISA5/c/KjAxE9YpyRR9v/iescxgAIHZAjFs7iwpJVkADx4ECQkBywSZNudFHqP/hwiZZFUxvADK/wXxxVQsox2hEVK8ol3mGD//ji+76bogdEIP0FW/RXbKjVhTSvIDkWmnYbDZsM5uR+vijbu2e/A+X4hMldJMbKZPH4ujho6pEITVEpHkBybXSYJ9rvu9Pv0Ufb/6HJSKyB4q+K6ab3YhNjEH0zT2xRaHHXrmoMZVdtQLKMRqR+vgjCO8c7mrz5X+ieutx5XIz3dwKw4SR2LF9h6JJVhalo1AgCEhytpm3or6uzi36gKf/CQ27nm5qhUpJVkCFqeyqFBCbNI3q1dOt3Zf/EYLSSVYuSopI0wKSw0DTSVMWX/6HJSra8/RGM3hYghpJVhdKiUjTApLD/5jNJoy8Z1Sr6OPL/8C5mXilqYVu9kinsFA1kqxuKJEvu6oExCZNU6c+Qnfx8j994/p4fCbIG2ySVanTG16Q9DWk0bqAJCUnO9stacqFt/8hQNWFGrrVI53CQjH4jnhkpCu/schB1nyZZgUk9R9dWlqKfXn7kDrVfeMQAvwPHL8Xvs4/Qjd7JWXKWDWSrG7I6Yc0KyCpN8XYpOmdo0fRXbz8D4tO4I1KbJJVhUc93JD69WTRrICkxFPSlAsf/8PS0vJrq8dafWGYMBLFp04rnt6gkSMKXRUC2mo2t0qacuHrfxKTElB3qd7jY61tcVNML8QOiEFGurKPetDIMZW1ewHZbDbkGnO8ikeI/2EZbhB+wjhl8lgc+eYwioqK6C5FkXoq06SApHyXOJKmxKN5BoDCrwp4+x+WEYZkusknbJJVbS8Eic/XaVJAUr5LtppNrZKmXEp//oW3//EXNZOsNFK9STUpIKlg0xZ00pRL5YVKXv5HCtRMstJI9SZt1wJamZnpMWnKkv95PpqvNPP2P5E99QCA+TOWo4E65swXw4SR2Ltnj2rpDS5SRKF2K6C8PXthLStrlTTlsn93niD/wx1XWuL5dIYvhhuSwJBflbqos02kiELtVkBGoyNt4S36AMCZ4tOi/I8+qjvdxBs2yapWlp7G3yjULgXEJk29bRyy+ON/xE5hcE5j9nq7qukNFn+jULsUkNlk8po0ZRHqf7hce921vBOqnugUForho5PUTrJKQrsTEJu2aGvlBRH+hyU0LBRhnTvBMGEk3SUILSRZpUBzAvJ3TmYvyPS288wi1v/0i7sZleWOEuD+wCZZP9qyie4KKDQnIH/mZJvNhu3btvn0PvDT/0jFCEOy0hd1esSfN63mBOQPOUYjwsPDfUYff/wPy67NeXSTYNiLOrWQZKXb+KIpAfnzTmCTpp4eV6UR63+4FB9v+4QqX9gkqxbSG2LQlID8eSc4zKj3pCkXsf6H5crlZsHPBHkjNjEG3fXdAnZFpikB+UOOMRv3Trzfa9KUiz/+Z2ByAmy19XSzXzwweYxmkqxC0YyA/Jm+2KTpJB7TlxT+B86HxKSCTbJqZXdaCJoRkD/Tl6+kKRcp/A8APDVb2pKmww1J2GbeqlqSVez5Mc0ISCyFBQWwlpW1uqbFG/76H5bu+m50k1+wZ8hUTLJenQLKzMjA4OTbcMutt9BdHvHH/8iJI8n6BzWnsatPQHyTpixS+J+Y2D6orqzBuRKrZCsxlhGGZM0kWfkS0AIym0zoF3dLm0lTLlL4H/ayzWOFFhwrtNDdfsGmNzIyVtBdmiVgBcQmTflsHLJI5X8AIG5AX7pJElKmjEWFtULt8/S80YSAxCzh+SZNuWjV/3Dprk41RNFoQkBCl/DcqoJ8kcL/0EiVzqBRoRqiaFQXkJjok2M0uqoK8kUK/wPOg/WV56v8fibIG2pUQxSLqgJiGGahmOjDJk35pC1YpPI/rACbGi5j8DDpohmNYcJIxashikFVAQkVD5ynLfgmTbkEgv/hMnhYgmbOkLWFqgISw8rMDIwcPUpQ9JHD/8DPB+v5wF7UqeUopJqAxHgfbxdk+kIq/0Pzzuvv002SwiZZzSbHBRBaRDUBiZm+jMZs3klTLlL5Hy5lP1cA4H/dnViGG5JUvajTF6oISEzmt7CgAKdO/sg7acpFav+TOCQeleVVCA27HlXn5RWQBpKsbaKKgMQk7tiqgnyTpixy+R9I/EyQN9hqiCYVam/wISAEVFRU1KqqIF/k8j9w7hrzKX3gLypVQ+SF4gISY55zsrPdqgoKQQ7/wzLCkKxIFGKTrEpXQ+SD4gISap59XZDpC6n9j1qoVQ3RF4oKSFT08VBVkC9y+Z+YuD642IZ5bmq8jNqqS24fLc38SiR4o7uzGuKKFcvpLlVRTECEkFSh0cdbVUG+yOV/wp3PBFVdqPF4wHDHB3swf8ZbWDJrletDiuV+yuSxmkuyKiYgocYZ7AWZxL2qoBDk9D9w1g7zlJE/V2LFn5+8F+kfLHB9RPb2X8RsklWNaoje0LSAcoxGwUlTLnL7n05hoXQTCCE4d7Ycv4vpTXdJwnBDkqbOkCkiIDHeh60qKDRpyiKX/6Ghi/BePF+Ny42XEX2zsN1yvmjpok4oJSCh3gecqoJio49c/gcA9D31qLvkOJ1KH+85d8YKnU6HzPnrsXTOGiydswaVFf5fB8NFzWqINLILSGz0EZM05SKn/4nsFYHmK47iu/NXzXLr+6XEiriBffHIsymuj+4R0p4hG2FIVrUaIhfZBSQm+pjNJlFJUy5y+x8W2gedO2tFXGIM+vbv4/oIuUb6l1ntaogs0v9lHMREH/asl5h9Hxal/I8nzp0pw00yGWgubJJV7fSGrAISE33YpKmYtAWLnP6Hpvh4iWuPx1Zbj0s1dYjqHYGW5hbHR8uv9JdIAlsNUe3aG7IKSCilpaX4prBQ9MqLRU7/A85G4n+LfsKpE2dw6vgZwLn/AwDzpi/D8w/Nw/MPzUNO+kduXyslWrioU2D9Pf6IqQjzypw5OHToa5j2fEx3CcKQNAov/M90jE25m+6SjPHJqZiUluJ6rDVl8lh6iCJsWGlCVZkNu/e23hEXik4ntB6jTBFIjPfxN2nKorT/keuEKl/UroYoi4DEeB9fVQX5oqT/YWmsF1bBUErYaojpK9S5Ik9yAYmJPo6zXka/xQMF/A8Xe10DomN6IjYxBscKLJIkTMWQMnksjh4+qko1RMkFJCb6sM/7+mueocD+zw9HLfjHdMd7pKbK5lgNDUtAp7BQpM9dh12b82Q/7kOjZjVESQUkJvqAR1VBvsjpf+x1drww5W/4cJ0Jj82YhMQh8bjISVHEJsa4dqVfm7XKY5ZeTtSqhiiZgMQcUwbPqoJ8kdP/hHUOw5xFz2NZ1mKvAu0UFoqUyWPx8tIZikchtZKskghIzMNiLEIuyPSFVP7HXmfHh1kmrEvPdWuPie3j9rk3uuu7uZ2bb7A3KiKoByaPUbwaoiQCEvOsD3hWFRSCv/7nvLUSH2aZ8PyUvwEA/jT5AXqIKI4VWPDarFU4ViDtjWY0g4clKF4N0W8BOX2PKAEZjdkYec8oSaKPFP7nk83/BgC8vektPDZ9kmRT4QhDMv4y9wnk7zyI9LnrZFutcashKhWF/BKQWN8DTtJUyBV1bSHG/5QU/wR7nd31+YyXp+Gx6ZNc9yC2RWQvPS5Y+T/nc1NML7y8dAaGG5KQv+Mg3S0ZbDVEpa7I80tAYsUD51kvf5OmXIT4H3Ypvm5FLurrGuhuXkT2ihD1mMYIQzImPZtCN0uGP9UQxayihb8CElBaWop9efskWXmx8PE/562VeCplpmspvixrsaCIJQfFx0uw9vX3JJ3WxCZZxQQE0QISo1YWMRdktgVf/xPeuRMWpP+9zaW40sQmxmDwsAQseXGlZJuQ7ElWMRuLQv+vogTkj/cpLS3lXVWQL/t35yEi0t3/eFqKh3UO470UV5IRhmQsNf4TjfVNSJ8r/J/uiRGGZFFJVp1Ot0jI7SmiBCRWPHAmTflUFRTCmeLT6HvrzYBzmlqXnutaij8+XdqiKCxh4WFoapAuidopLBSTnk3Bv1bNprtE4Wc1RPkEJDTEcRFSVVAIlRcqXVPSJ5v/jcieetdSnM+KSgwxcX0kmW7aoupCDfJ3HhT9c/yohpjK9/8sWED+RB8hVQX58mHOBw7/k+wQ0IyXp+FPU+6XTThKwl5gJXYT0p9qiHynMkEC4qtKbwipKuiLL/d/gRenzcRHuR8itFOoJr2Nv7DTGncTUih+VkOUVkD+Rh++VQXb4vSPpzFp3J9h3vgRps1MQ1h4Jwwaym//J1BhNyHFPDbrZzVEn1MZbwH5+ka+kCpp2rlLOFbnrsXq3LW4begQN/+jBlWV0u3f+CI2Mcbtc77lpgwTRoquhqjT6Ra19b/nLSB/og9bVVBo0rS+rh65a7Oxe8enrraoXj1dInTt/zj9j5Kwoj12SLg3kQpT9i5eubXhhiS/Lups63/PS0BtKZAPbFVBvtGnwlqOpfOX4OmHpwLwXhNj/+48dArr1C79Dx/Y3JqvTUgpqiF604BPAfmzaQgRVQUBYPWbK3Hb0CHI2boR02ameTXdZ4pPqxJ9tAS7CdkpLBSNdu/7UmySVWh6g8XbVOZTQHzEU1NdjdraWroZcFYV9JU0Pf3jabfPl65ehvt4rNbU9j9agX2Mg74phIsryeqjGmJjYyPOnTuH6qpqusujiNoUED3YE3V1dbhn1F148fkX6C7XWS9vSdPdOz7Fi9NmYs2yTLrLJ2r6n0BgbtqbraY1thqityi0b28ebksciKefeBLjx41D8m1DsH7dOjAM4xpDi8irgPhOXe9v2ID4hARUlJfTXV6Tpt8d/haTxv0ZX+Z/gWkz07A6d61bPx+04H9iYvuI2uBTgvmrZqGxvsltE7K7sxriR1s20cMBABaLBaMNBuQd2I+CI4fx/ocfYON772NlhvsbnCsirwLiI54Gux0bcnMx4y/PoaLCUTuCxWazeU2a9uzdE6tz12Lp6mVtTm1toQX/ExbeiW7SDPQmJHtKpK1qiEUWC+IT4l2fx8fHY/acOXgv1/3ZcHD04VFAfKYuAPhg40YMHXo7Bg4cCLvdDrv9t6f7coxGhIeHY8jtSchdm40v93/h6uMuxcUS9D/8YDch2T2ktqohWiwn0D/+NwEBQP+EeNTX16OystKtHc7DFK0ExHfqampqgnF9Nma+8Dy63XgjOnbsiHLnNGaz2WBcn42o3j3x1IOOpbjYSOOJoP/xj+tDr2tVDbG2pgbl1nIkJLi/prU1jj2m6667zq3dibuAhBzP+WjTJvzu979HREQEzldU4Mbu3XHeOY2dLCpCg92OQUmDsftQXptLcTFowf8EKg32RtzQvQsA4ONt213tFosFN3Trhqie7jPDmf+egV6vR5cujq+haBWBfCbPAODKlSvIWpeF6upqTJ3yGKZOeQy1NTU4X3EeAHDHsGEYM3YM9uzcjQpra3PtL1rwP3A+WF9bLTw9oBbs8aKjB09g/oIF+OusF119RRYL4qnpCwA+2b4dY/7oPQcnSkBbTWZEREQg//8OYO/+fOzdn487hg1zTWEA8FZGBnrfFI25L/4D9XWOG02lQiv+J7JXBFqa5bmBTErOlViRPncd3ln6PkbdeQ8OHjqEac887TbGYrG0mr62b92GoqIiPDdzpls7F5eA+BrnlpYWvLt2Lf7yvPs3jYyMxPnzv63EunTpgqzsbFRYK7B03hK3sf4Q9D/8abA3wrR+F16bvQpovgabtnyEFRkZHqejk5Yi9E+IB8Mw+OnsWaxYvhwL5s9H+spM9O7t/c5Hl4D4ep9Ptn+Mjtd1xB/HjXNrj4yKck1hLNHR0cjKXo+DB75A7tpstz6xBP0PP77OP4K5aW+i4MB3WL5iBXbvzcMdw4bRwwAAly9fhrW8HH97+RUMShiAp6Y+gYryCny+Lw/3jR9PD3dDBwErL7FsM2/F3195Ba+vWoY7R4+iuwXx6H0P4Xd9e2NB+t/pLsX5MMuE3R/vwxvGf9JdqlF8vAS7Nueh+EQJnpj2JF6a87LHiOOJ5uZmhOh0uKZDB7rLKyFyiwcAHkp9GA8+9BDemL+kVd5LKFrxPyyNXjLgStNgb8SGlSakz1uHzqHdsGv3Z1i4aDFv8QDAtddeK0g8ABAit3hY3spIR3R0NObN+rtoU601/5OYlIBGCU9miCV/50HMTXsTZ0+WYvmKFTBt3eZxRSUH9CpMVjabTKivs2Pui+Kmn6D/caf4eAmWzFoJU/YuPJP2LHbvzcNDqQ/Tw2SDELJIUQF16dIFm01b8N9Tp7F0vvCVmVb2f9Sm6kIN1r7+HtLnrUNURDT+89VBQV5HKkJCQhYrKiA4E3QLFi7C5zs+c3tUlQ9a8z9qsGtzHpbMWoXz56rxblYWtpjNiI6OpocpgRkAQgghinggLg+lPownn56GN+a/xttUa83/KE3x8RLMTXsTB3YdwjNpafjqUAHGjvsjPUxJHAIKCQlZrIaIFixcCMPYMZg1bSavdIcW/U9f5++Sv1O++36qLtQgfe46pM9bh0EDb8Nne/bgpTkv08MUhRCySKfTOQQE51ymhohWCEh3aNH/yHn6tcHeiF2b8zA37U3UVTdi05aPsN6Yo9Z0xWIGMCkkJGQx2+DyQCEhIYt1Op1OSSF16dIFKzIyUGGtwGofj7VeTf7n6/wjeG3WKhzYdQjzFyzAwUOHvO4iK4Ez4uh0Ot0kNvKwtDLRSkej+Ph4ZGWvx+c7PvOa7rha/A+b9Hxvldlr0lNpCCGLuBGHppWAwIlGSkWkO4YNw/wFC5D7jtHtyUWW/bvzEBrWPs+/Q2DSUyHMACbpdDpdW+KBNwFx4YqpLfwV2rRnnnakO+a1TncUnzzF+/5DpYmI7NGqcrMQhCQ9lcA5XbWaqrzhU0B8kSJq/WvRQkTfFI035i9xM9VVFy9q1v9E9dajoV54PozdRX5vlRmpkx7BwUOHFN1FpuAdcWgkExAXob8Ei2On2oQKa4Ur3ZH/eT5amlvajf/hJj0je/QSlfSUirbMMV9kEZA/sOmOY0e+w+plK9uV/3EkPZfh7MlSvJuVpWjSk4UjGsHRxhOyCUjsNAbnymz5ihXY+sEWfH/0W836H76wu8iOpGcadu/NU3IX2TU9SSUaLoJrZApBTN1ULq8uXowPN25E/8RY1xTG3gXdN7aPrBt5fHn15eU491MpFqxpfTlm1YUabFm/E98XFiH59qFIz8xUaiPQDMAsdloSgtwCMvF9UN8bb725DCd/LILdbkdZaRnKqbSHvmcEIntGIKxzGPrGOaY5JUXm7anEXZvzkL/zK3Tt2hUZmStlX1n52q+RC7kFlArARLf7i81mw0lneceCQ44juidO/ACbzYaysjKUlyknMlpAxwosMGXvQlPDFTyT9ixmvdQ6MolFLZG0hawCgsBprKWlBZUXKhESooM+MhIiqlC78CQyi+U4Ll26hDqbDad+LHYbL1ZkrIBeeWMGNqw0ofhECUaPGY2Fi1/lPV3p/PlDVUb2X5yvgGw2G5IGDUa/W27BhQsX0LlzZ6xaswaDBg+ih0oKe8lAkaUINpsNv5z7GaXnzvEW2XlrJQq/PIK6S/WIjbsFi15dImi6CmTxQEsCKiwoxMznnsPRY9+BYRjMmT0b1jIrTNs832WjJL5E1qVrZ7w46yXBeSstTkmag/AkJ9tIHp88xfX5hpwckjL+frcxXOifw0IISSWEmOjxSlNTXU2OfXeMWMusdJcL+ncPRGTbBxJKUdFvZ7N/OnsWH7y/EdNnzKCH+USn05mdO6suAExin6CTmyOHDyPlvvEYOXwEXnnpJYwdPRpnS87Sw4LwhX7XeWP8uHFk+NDbyd13jiKDByQSi8VCD3GD/jl8YRhmIf29pGKbeSsZlDCAbDWZSUtLCyGEkKamJnqYC/p3C+IB+kXzxOXLl0lsTF9y7Ltj5OiRI2TOrNnkwYkTCcMw9FAX9M8Rg5Ri+uWXX0hc335k146ddJdH+N5FcNVDv3CeOHH8OEnsH+8STEtzM+l/Syz54fvv6aEu6J8jFqlE9K9588hjj06mm71C/x6BiiY8kMViwa39+7v2fertdly5cgUdO3akhwJ+5tlopFoFHcjfj7vuuZtubvdoQkAnLUUuA11uLcecWbORlJyM2Lg4eigg4T+dxV9BNjc3w2q1Ii7uVrrLI/7+PC2hCQGdOXMGmzdtwqCEAXhw4kT069cPWcZsjzvRcrz4/j4HzjAMdDodOnS4hu7yiNRvgHYNPfd7o6GhgTQ3N9PNLpQwnfTPFMJdfxhJ1qxaRTd7hP65QdqAfvHEoIR44Keh3pCTQwYlDCD78vJIc3Mzqa+vJyVnSuhhhAQFJAz6xROKUuJhESuiX3/9lfzv6tVk8IBEEte3H+n7+z7krzOfp4cp/vcEPPQLyAe1X2R/UyEVFRWkvr6ebiYkGH2EI+QdrbZwWJz5NMnRyt8XcPAQkYlPhWAl4fE7C4b+GUEE4OkfEgjvSPp35ksg/G1BFMKT+D1xNYrm/wHHHDCD6/SS6AAAAABJRU5ErkJggg==[/img]Essa é a fórmula de cálculo do volume de uma pirâmide triangular. Agora, vamos mostrar que ela também é válida para uma pirâmide qualquer. Para isso, considere uma pirâmide de altura medindo h cuja base é um polígono convexo de n lados com área [math]A_b[/math]. A figura mostra um exemplo para n igual a 5. Como o polígono é convexo, a partir de um dos vértices da base traçamos todas as [math](n−2)[/math]diagonais do polígono da base, obtendo [math](n−2)[/math] triângulos. Podemos dividir a pirâmide em [math](n−2)[/math]pirâmides triangulares de mesma altura da pirâmide original e área da base [math]A_1,A_2,...,A_n_{-2}[/math], traçando planos determinados por essas diagonais e pelo vértice [math]V.[/math] O volume [math]V[/math] da pirâmide será igual à soma dos volumes das [math](n−2)[/math]pirâmides triangulares. Então, pelo resultado anterior:[/justify][br][math]V=\frac{1}{3}.A_1.h+\frac{1}{3}.A_2.h+.........[/math] [math]+3.A_n_{−2}.h\Longrightarrow[/math] [math]V=\frac{1}{3}.h(A_1+A_2+...+A_{n−2})[/math][br][br]Mas [math]A_b=A_1+A_2+...+A_{n−2}[/math]. Então, o volume de uma pirâmide qualquer de altura medindo h e área da base [math]A_b[/math] é igual a:[br][br][math]V=\frac{1}{3}A_b.h[/math]

[justify]Em uma feira de artesanato, foi construída uma tenda com tecido no formato de uma pirâmide hexagonal regular com [math]8m[/math]de altura e aresta da base medindo [math]4\sqrt{3}m[/math]. Considerando que quem armou a tenda deixou uma das faces laterais como porta (sem fechamento do tecido), calcule a quantidade de tecido necessária para a cobertura da tenda. [br][br][b]Resolução:[/b] Primeiro vamos representar a tenda e sua base:[/justify]

No triângulo [math]AOB[/math], [math]m[/math]é a medida do apótema da base, então:[br][br] [math]m=\frac{l\sqrt{3}}{2}\Longrightarrow[/math] [math]m=\frac{4\sqrt{3}.\sqrt{3}}{2}=6\Longrightarrow m=6[/math] [br][br]Aplicando o teorema de Pitágoras no triângulo [math]VOM[/math], temos:[br][br][math]g^2=6^2+8^2\Longrightarrow g^2=36+64=100\Longrightarrow g=10[/math] pois [math]g>0[/math][br][br]Cálculo de área [math]A_f[/math] de uma face da pirâmide:[br][br][math]A_f=\frac{l.g}{2}\Longrightarrow A_f=\frac{4\sqrt{3}.10}{2}=20\sqrt{3}\Longrightarrow A_f=20\sqrt{3}[/math][br] [br]Como uma das faces laterais não usará tecido (porta), a área lateral será dada por:[br][br][math]5.20\sqrt{3}m^2=100\sqrt{3}m^2[/math][br][br]Portanto, serão necessários [math]100\sqrt{ }3m^2[/math]de tecido. Se resolvemos essa expressão, teremos um resultado mais preciso, veja: [br][br] [math]100.\sqrt{3}=100.1.7320=173,20m^2[/math] de tecido.

Calcule o volume de uma pirâmide cuja base é um quadrado de 3 cm de lado e sua altura mede 10 cm.[br][br]Observe a pirâmide da figura.[br][br][img]data:image/png;base64,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[/img]Como [math]V=\frac{1.A_b.h}{3}[/math], precisamos determinar a área da base [math](S_b).[/math] A base é um quadrado, logo:[br][br][math]A_b=l^2\Longrightarrow A_b=3^2=9\Longrightarrow A_b=9cm^2[/math]. [br][br]Cálculo do volume (V):[br][br][math]V=\frac{1.A_b.h}{3}\Longrightarrow V=\frac{1.9.10}{3}=30\Longrightarrow V=30cm^3[/math][br][br]Portanto, o volume da pirâmide é [math]30cm^3[/math]

1-Em uma pirâmide quadrangular regular, cada aresta lateral mede 15 cm e cada aresta da base mede [br]18 cm. [br][br][br]Calcule:

2- (UFPE) Uma pirâmide hexagonal regular tem a medida da área da base igual à metade da área lateral. Se a altura da pirâmide mede 6 cm, assinale o inteiro mais próximo do volume da pirâmide, em [math]cm^3[/math].