Tuesday, July 17, 2012

Express the area A of a circle as a function of its circumference C

For a circle the circumference can be expressed in terms
of its radius as 2*pi*r


The area of a circle can be
expressed in terms of its radius as pi*r^2


Now we have C=
2*pi*r and A = pi*r^2


From C= 2*pi*r we can derive r = C /
(pi*2) by dividing both sides by pi*2.


And from A = pi*r^2
we can derive r= sqrt ( A / pi).


Now equate both the
expressions we have for r


=> C / (pi*2) = sqrt ( A /
pi)


Square both the
sides:


=> C^2/ (pi*2)^2 = A /
pi


=> A = (C^2 * pi) / ( pi^2 * 2^2
)


=> A = C^2/ pi*
2^2


=> A = C^2 /
4*pi


Therefore we have Area = C^2 / 4*
pi

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