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

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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**

We know:

Area of a circle = A = (pi)r^2

Circumference of a circle = C = 2(pi)r

Where pi is a constant and r is the radius of the circle.

Using these two formulas we can express A in terms of C as follows:

C^2 = [2(pi)r]^2

==> C^2 = 4[(pi)^2]r^2

==> C^2 = 4(pi)[(pi)r^2]

As (pi)r^2 = A

==> C^2 = 4(pi)A

Therefore:

A = (C^2)/[4(pi)]

We know that the area of the circle is given as the following function:

A = (r^2)*pi where r is the radius.

But we know that:

C = 2r ( c is the circumference)

==> r = C/2

Now substitute with r = C/2

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

= C^2 / 4)*pi

= (C^2)*pi /4

**A = (c^2)*pi/4**

Area of a circle to be expressed in terms of its circumference.

The area A of the circle = pr^2.........(1), where p has the value of pi and r is the radius of the circle.

The circumference C of the circle is given by: C = 2pr.........(2).

Elimnate r between the equations (1) and (2).From (2) we get r = C/2p .Now by substitution r = C/2p in eq (1), we get:

A = pr^2 = P (C/2p)^2

A = P(C/2p)^2

A = PC/4p^2

A = C/4p is the area A of a circle whose circumference is C.

A = C^2 / 4*pi

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