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circles

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If an arc of 60° on circle  has the same length as an arc of 45° on circle , what is the ratio of the area of circle  to the area of circle ?

Feb 17, 2021

#1
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Let  A  be the  arc length

A   =  R1  ( pi/3)    →     A / (pi/3)   =  R1

A  =   R2  ( pi/4)   →     A / (pi  / 4) =  R2

R1  / R2   =   [A / ( pi/3) ]  /  [ A /(pi/4) ]  =   [ 3/pi ]  / [ 4/pi ]  =   3  /  4

Ratio of areas =  (3/4)^2   = 9/16

Feb 17, 2021
edited by CPhill  Feb 18, 2021
#2
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If an arc of 60° on circle A  has the same length as an arc of 45° on circle B, what is the ratio of the area of circle A to the area of circle B?

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Circle M radius       R(M) = 1                     |       Circle N radius   R(N) = [(L(A) * 8)/pi] / 2 = 4/3

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Arc length   L(M) = 2rpi = 1.047197551     |        Arc length  L(N) = L(M)

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Circle M  area          A = 3.141592654      |       Circle N area      A = (4/3)2pi = 5.585053606

The ratio of the area of a circle M to the area of a circle N is:          A / B = 16

Feb 18, 2021
edited by jugoslav  Feb 18, 2021
edited by jugoslav  Feb 18, 2021