Question

A circle can be cut into smaller and smaller slices and rearranged in the shape of a parallelogram as shown.



The formula for finding the area, A, of a parallelogram with base b and height h is shown.


A = b x h
Which equation represents the circumference of a circle in terms of its area?

Answer 1

To find an equation that represents the circumference of a circle in terms of its area, we start with the formulas for the area and the circumference of a circle.

1. The area \( A \) of a circle with radius \( r \) is given by:
\[
A = \pi r^2
\]

2. The circumference \( C \) of a circle is given by:
\[
C = 2\pi r
\]

Now, we want to express the circumference \( C \) in terms of the area \( A \).

We can solve the area equation for \( r \):
\[
A = \pi r^2 \implies r^2 = \frac{A}{\pi} \implies r = \sqrt{\frac{A}{\pi}}
\]

Next, we substitute this expression for \( r \) into the circumference formula:
\[
C = 2\pi r = 2\pi \left(\sqrt{\frac{A}{\pi}}\right)
\]

Now, simplifying this expression:
\[
C = 2\pi \sqrt{\frac{A}{\pi}} = 2\sqrt{A\pi}
\]

Thus, the equation that represents the circumference of a circle in terms of its area is:
\[
C = 2\sqrt{A\pi}
\]