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# Help quick

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A chord of length $6$ units divides a circle into two distinct areas. If the circle has a radius of 6 units, what is the area of the larger region, in square units? Express your answer in simplest radical form in terms of $\pi$.

Nov 7, 2019

#1
+19724
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Nope, still not correct     Ooops....  back to the drawing board......

Nov 7, 2019
edited by ElectricPavlov  Nov 7, 2019
edited by ElectricPavlov  Nov 7, 2019
edited by ElectricPavlov  Nov 7, 2019
#2
+19724
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Draw a line from the center to each of the endpoints of the chord

this forms a equilateral triangle     the central angle is 60 degrees   (pi/3 radians)

the area of this sector is   60/360 of the TOTAL area of the circle

pi r^2  x  60/360   = 6 pi

Now subtract the area of the equilateral triangle to get the area enclosed by the chord

area of equalateral triangle = r^2 (sqrt3) / 4   = 9 sqrt3

6pi - 9 sqrt3 = area of chord

Total area of cicle - area enclosed by chord = larger area

36 pi      -    ( 6pi  - 9 sqrt3)

30 pi + 9 sqrt3     units2

(there has to be a better way! )

Nov 7, 2019
#3
+105370
+1

The smaller area  is the  difference between the area of a sector  of 1/6 of the circle  less the area of a equilateral triangle with a side of 6.....this is given by

[(1/6) pi * 6^2   -  (1/2)6^2 * √3/2  ]    (1)

So....the   larger area  =

Area  of circle  - (1)

So

pi*6^2  -   [(1/6) pi * 6^2   -  (1/2)6^2 * √3/2  ]  =

(5/6) pi * 6^2  + 9√3   =

[30pi + 9√3] units^2

Nov 7, 2019
#4
+19724
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Wel-l-l-l  ,  I guess there WASN'T an easier way !

ElectricPavlov  Nov 7, 2019
#5
+105370
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I don't see one, EP......

CPhill  Nov 7, 2019
#6
+19724
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I felt like , if anyone knew one, YOUR answer would 'learn it to me'    Haha....

ElectricPavlov  Nov 7, 2019
#7
+105370
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LOL!!!.......

CPhill  Nov 7, 2019