Question

he formula for the area of a circle is A = πr
2
. This formula allows you
to find the area of a circle if you know the radius or diameter. How can
you find the area if you know only the circumference?

Answer 1

Answer:

d/2 = radius

Step-by-step explanation:

Circumference = pi*diameter

C/pi = diameter

Diameter/ 2 = radius

Answer 2

Answer:

$\text{We have the formula for the circumference of a circle with radius }r:\\C=2\pi r\\\text{In case we know the diameter of the circle, the circumference will be: }\\C=\pi d\\\text{Now, according to the question, we know the circumference of the circle. So}\\\text{first let us write the formula for the circumference:}\\C=2\pi r\\\text{or, }r=\dfrac{C}{2\pi}$

$\text{Now we write the formula for area of a circle with radius }r:\\A=\pi r^2\\\text{We now can substitute the value of }r\text{ from the above equation:}\\A=\pi\bigg(\dfrac{C}{2\pi}\bigg)^2\\\\\text{or, }A=\dfrac{C^2}{4\pi}$

$\text{Using this relation, you may find the area of a circle when you know only the}\\\text{circumference.}$

In case you want to proceed with the circumference formula involving diameter,

$C=\pi d\\\\\text{or, }d=\dfrac{C}{\pi}\\\\\text{Now let's write the relation between area and diameter of a circle.}\\\\A=\pi\dfrac{d^2}{4}\\\\\text{Let's substitute the value of }d\text{ from above relation,}\\\\A=\pi\dfrac{\bigg(\dfrac{C}{\pi}\bigg)^2}{4}$

$\text{or, }A=\dfrac{\pi C^2}{4\pi^2}\\\\\text{or, }A=\dfrac{C^2}{4\pi}$