Octógono

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Octógono
Octógono regular
Tipo	Polígono regular
Arestas e Vértices	8
Símbolo de Schläfli	{8} t{4}
Diagrama de Coxeter
Grupo de simetria	Simetria diédrica (D₈)
Área	$2(1+{\sqrt {2}})a^{2}$ $\simeq 4.828427a^{2}$ (sendo a = comprimento da aresta)
Ângulo interno (graus)	135°
Propriedades	convexo, cíclico, equilateral, isogonal

Em geometria, o octógono é um polígono com oito lados (e portanto oito ângulos internos, oito vértices e oito ângulos externos).^[1]

Octógono regular

Um octógono regular tem todos os lados de mesmo tamanho e todos os ângulos com a mesma medida.

Construção de um octógono regular com régua e compasso

Área de um octógono regular

A área de um octógono regular de lado 'a' é $A=2a^{2}\cot {\frac {\pi }{8}}=2a^{2}({\sqrt {2}}+1)\simeq 4.82843a^{2}.$

Sabendo o comprimento 'm' do apótema, e considerando o octógono composto por 8 triângulos isósceles, podemos recorrer a uma fórmula mais simples

$A=8\times ({\frac {a\times m}{2}}).$

Medida dos ângulos internos

$ai={\frac {(8-\ 2).\ 180}{8}}$

Logo:

$ai={\frac {6.\ 180}{8}}$

Então:

$ai={\frac {1080}{8}}=135$

Daí conclui-se que a medida do ângulo interno de um octógono regular é 135.

Soma dos ângulos internos

$(n-\ 2).\ 180\rightarrow (8-\ 2).\ 180=\ 6.\ 180=\ 1080$

Daí conclui-se que a soma dos ângulos internos de um octógono regular é 1080.

Medidas dos ângulos externos

$ae={\frac {Se}{n}}$

Logo:

$ae={\frac {360}{8}}=45$

Daí conclui-se que a medida do ângulo externo de um octógono regular é 45.

Medida do ângulo central

$ac={\frac {360}{n}}$

Então:

$ac={\frac {360}{8}}=45$

Assim, conclui-se que a medida do ângulo central de um octógono regular é 45.

Número de diagonais

$d={\frac {n.\ (n-\ 3)}{2}}$

Então:

$d={\frac {8.\ (8-\ 3)}{2}}$

Logo:

$d={\frac {8.\ 5}{2}}$

Então:

$d={\frac {40}{2}}=20$

Então, conclui-se que o número de diagonais de um octógono regular é 20.

Ver também

Lista de construções do desenho geométrico

Referências

↑ «significados.com.br - Significado de Octógono». www.significados.com.br

v d e Polígonos
Convexos	Triângulo · Quadrilátero · Heptágono · Octógono · Eneágono · Heptadecágono · Octodecágono · Hendecoságono · Docoságono · Pentacoságono · Triacontágono · Tetracontágono · Quiliágono · 65537-gono
Estrelas	Pentagrama · Hexagrama · Octograma

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