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Area of circumscribed circle

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by maihuna » Mon Aug 10, 2009 8:32 am
For any given regular polygon:

a = 2R sin(pi/n)

Here 10 = 2R sin60 = 2R*\/3/2 => R = 10/\/3

so area = PI R^2 = PI* 100/3 = 100PI/3
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by madhur_ahuja » Mon Aug 10, 2009 8:35 am
maihuna wrote:For any given regular polygon:

a = 2R sin(pi/n)

Here 10 = 2R sin60 = 2R*\/3/2 => R = 10/\/3

so area = PI R^2 = PI* 100/3 = 100PI/3
Can you elaborate ? What does a and n signify here?
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by maihuna » Mon Aug 10, 2009 8:38 am
a will be side of polygon
n will be no of sides.
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by gmatv09 » Wed Aug 12, 2009 8:45 am
Radius of a circle circumscribing a equilateral triangle (R) = a/sqrt(3)
[Formula to remember]

R = 10/sqrt(3)

Therefore area of the circle = 100/3 * pi
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by Svedankae » Wed Aug 12, 2009 9:00 am
gmatv09 wrote:Radius of a circle circumscribing a equilateral triangle (R) = a/sqrt(3)
[Formula to remember]

R = 10/sqrt(3)

Therefore area of the circle = 100/3 * pi

So by that logic is the radius of a circle circumscribing a regular pentagon (R) = a/sqrt(5)

With "a" being the length of one side of the pentagon?
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by gmatv09 » Wed Aug 12, 2009 11:16 am
No. this formula is applicable to triangles only - AFAIK
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by maihuna » Sat Aug 15, 2009 9:15 am
gmatv09 wrote:No. this formula is applicable to triangles only - AFAIK
:)
Actually this formula is applicable for all cases.
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by gmatv09 » Sat Aug 15, 2009 12:05 pm
thanks !
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by maihuna » Sat Aug 15, 2009 12:21 pm
Svedankae wrote:
gmatv09 wrote:Radius of a circle circumscribing a equilateral triangle (R) = a/sqrt(3)
[Formula to remember]

R = 10/sqrt(3)

Therefore area of the circle = 100/3 * pi

So by that logic is the radius of a circle circumscribing a regular pentagon (R) = a/sqrt(5)

With "a" being the length of one side of the pentagon?
Note this formula (yes applicable only for regular polygons)

a = 2rtan(PI/n) = 2Rsin(PI/n) where r and R has obvios meaning

for pentagon: a = 2Rsin(180/5) = 2Rsin(36) you know finding 36 will not be possible w/o a sine table, so its normally useful for 30,45,60,90 when PI/n results in such scenario.
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