How do you write the area a of a circle as a function of its circumference?

Question

James Allen · Accepted Answer

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)]#

Elijah Allen · Answer

undefined To write the area (A) of a circle as a function of its circumference (C), we can use the relationship between the circumference and the radius of the circle. The circumference (C) of a circle is given by the formula:

C = 2πr

Where r represents the radius of the circle. From this formula, we can express the radius (r) in terms of the circumference (C):

r = C / (2π)

Once we have the radius expressed in terms of the circumference, we can use the formula for the area (A) of a circle, which is:

A = πr^2

Substituting the expression for the radius (r) in terms of the circumference (C) into the formula for the area (A), we get:

A = π(C / (2π))^2

Simplifying this expression further, we have:

A = π(C^2) / (4π^2)

And finally, simplifying it to express the area as a function of the circumference:

A = C^2 / (4π)