Problema (nível 4): Na figura seguinte, todos os arcos são arcos de circunferências de raio 1. A circunferência central passa pelos pontos de contacto (tangência) dos arcos de circunferência periféricos. Quanto mede a área da região sombreada? (image1)


Conteúdo referenciado:
[image1]
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Resolução: Solução 1: Constata-se imediatamente pela figura abaixo que a (medida da) área da região sombreada é dada por $Q-12F$, onde $Q$ designa o valor da (medida da) área de um quadrado de lado $4$ e $F$ representa a quarta parte do valor da diferença entre a (medida da) área de um quadrado de lado $2$ e a (medida da) área do círculo nele inscrito. Portanto, como $Q=16$ e $F=\frac14(4-\pi)$, o resultado pretendido é $4+3\pi$. (image2) Solução 2: Alternativamente, considerando a figura (image3) constata-se imediatamente que a (medida da) área da região sombreada é dada por $Q'+4F'$, onde $Q'$ designa o valor da (medida da) área de um quadrado de lado $2$ e $F'$ representa três quartos da (medida da) área do círculo de raio 1. Portanto, como $Q'=4$ e $F'=\frac34 \pi$, o resultado pretendido é $4+3\pi$.


Conteúdo referenciado:
[image2]
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