Consider the circle of radius r pictured below with central angle , measured in radians, and subtended arc of length s. Prove that the area of the shaded sector is A = 1 2 r 2 θ.Let's say the entire circle has area B = π r 2 . Using proportionality arguments, the area of a portion of a circle is then θ 2 π B = θ 2 π π r 2 which reduces to A = 1 2 r 2 θ.Is that right? It seems too easy for a prove. I got this question from a online book which doesn't contain the answer, so I can only resort to stack exchange.

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Consider the circle of radius r pictured below with central angle , measured in radians, and subtended arc of length s. Prove that the area of the shaded sector is   A  =      1    2        r    2    θ.Let&#039;s say the entire circle has area   B  =  π      r    2  . Using proportionality arguments, the area of a portion of a circle is then       θ          2      π        B  =      θ          2      π        π      r    2   which reduces to   A  =      1    2        r    2    θ.Is that right? It seems too easy for a prove. I got this question from a online book which doesn&#039;t contain the answer, so I can only resort to stack exchange.

Sasha Pacheco · Accepted Answer

A  =  ∫  ∫  d  x  d  y ?Let us use polar coordinates.  x  =  r  c  o  s  (  t  ) and   y  =  r  s  i  n  (  t  )The Jacobian is   r  d  r  d  t.thus  A  =      ∫    0    θ        ∫    0    R    r  d  r  d  t  =  θ            R      2        2

Consider the circle of radius r pictured below with central angle , measured in radians, and subtend

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