Could someone explain to me why the following holds? Area of sector Area of circle = θ 2 π When deriving the area of a sector my book just quotes the above but doesn't explain why it holds. Could someone explain?Also, a similar argument is used when deriving arc length, which I don't understand.By the way, I know that θ is the angle subtended by the arc and 2 π is the angle in a full circle.How does the ratio of the areas make it equivalent to the ratio of their angles?

Question

Could someone explain to me why the following holds?      Area of sector    Area  of  circle    =      θ          2      π      When deriving the area of a sector my book just quotes the above but doesn&#039;t explain why it holds. Could someone explain?Also, a similar argument is used when deriving arc length, which I don&#039;t understand.By the way, I know that   θ is the angle subtended by the arc and   2  π is the angle in a full circle.How does the ratio of the areas make it equivalent to the ratio of their angles?

Isabella Rocha · Accepted Answer

This is more intuitive than axiomatic. If you cut a pie into   n equal pieces then each piece will have the following properties.1/ The central angle   θ will be   θ  =            2      π        n  2. The arc length   L of the each sector will be the same, i.e.   L  =            2      π      r        n  3. The area   A of each piece will be the same, i.e.   A  =            π              r        2              n  Thus, the quantities angle, area and length will be       1    n   of   2  π, the area of the circle and the circumference of the circle, respectively.Using this we can derive the usual formulas:  A  =            π              r        2              n    =  π      r    2        θ          2      π        =                    r        2            θ        2    L  =            2      π      r        n    =  2  π  r      θ          2      π        =  r  θ

cortejosni · Accepted Answer

How about some calculus? This is the area of the sector of radius   r and angle   θ:      A          s      e      c      t      o      r        =      ∫                  r        ′            =      0              r            ∫                  θ        ′            =      0              θ            r    ′    d      r    ′    d      θ    ′    =      r    2        /    2  ⋅  θ  .For the full circle, integrate   θ from 0 to   2  π; the ratio of the two answers will get you the relation in your question.

Could someone explain to me why the following holds? (Area of sector)/(Area of circle)=theta/(2pi) When deriving the area of a sector my book just quotes the above but doesn't explain why it holds. Could someone explain?

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