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