Question

Find the diameter of the circle that has a circumference equal to its area.

2
4
8
10

Answer 1

The formula for circumference of a circle:

`C=2 \pi r`

The formula for area of a circle:

`A=\pi r^2`
r - radius

The circumference is equal to the area.

`C=A \\ 2 \pi r=\pi r^2 \ \ \ |/ \pi r, \ r>0 \\ 2=r \\ r=2`

The diameter is two times the radius.

`d=2r=2 * 2=4 \\ \\ \boxed{d=4}`

Answer 2

Circumference may have the same numerical magnitude as area, but they will never be "equal". Circumference is a length or distance, and area is an area.
So they can never have the same units, and quantities can't be equal without
the same units.

You might say that π D (circumference) = π (D/2)² (area) .

Then D = (D/2)²

D = D² / 4

1 = D / 4

D = 4 .

But this is no more than a parlor trick. In order to do it, you need to
ignore units completely, and that's unacceptable in math.