The area and the circumference of a circle are the same..what is the radius of the circle?

Question

Also, what is unique about this radius?

What is unique about this radius?

Answer 1

NAIVELY, and only APPARENTLY, r = 2. Why? :

Because then π r^2 = 2 π r, and therefore, dividing by the product (π*r), r = 2.

Notice that this question really makes an assumption that UNITS DON'T MATTER, which is NOT true. In fact:

The dimensions of area are L^2, whereas the circumference is only of dimension L. This should be a warning sign : in any VALID equation with a definite and unique physical answer to it, the dimensions MUST balance. Otherwise, any conclusion is not determinate and absolute, but rather is frame- or unit-dependent. This means that a choice of length can make this property true for ANY given circle, provided that the UNIT OF LENGTH is ONE HALF THE RADIUS.

So basically, it's a DAMNED SILLY QUESTION (a technical term often employed by we in the physico-mathematical trade).

As I have just pointed out, dimensional considerations mean that NOTHING, absolutely NOTHING, is unique about this radius. But if you really mean "What is unique about this number 2," one thing is that, apart from 0 (that is, zero), it's the only number whose sum with itself is equal to its square.

Live long and prosper.

Answer 2

In a given circle, let 'r' be the radius. If the area and circumference are the same, then pi (r^2) = 2(pi)r
so the radius is 2 because when you divide each
side of the equation by 'pi' you are left with r^2 = 2r
therefore r=2.
Unique about this radius? 2 is the lowest prime number -
if that's what you mean.

Answer 3

Area of a circle = pi r ^2
Circumference = 2 pi r

So pi r^2 = 2 pi r. Dividing each side by pi r gives r = 2. So area and radius are both 4 pi.

Answer 4

A=pi r r and
C=2 pi r.

Pull out a pi and an r and you have r=2.

Nothing is notably unique about this radius. More information is needed to give a correct answer on why it's unique.

Answer 5

the area of a circle is the amount of square units inside the circle.

the circumference of a circle is the distance around the circle.

the radius is half of the diameter, which is the distance across the circle.

if this is for your homework, figure it out urself.

Answer 6

If Circumference = Area, then 2 pi r = p r square factor: 2 pi r - pi r square pi r (2 - r) = 0 solve for r: r = 0 or r = 2 r = 0 does not exist, so r must = 2

Answer 7

(pi) r^2 = 2 (pi) r
r = 2

unique in the sense that its only @ r = 2 where the area and the perimeter of a circle are the equal.

Answer 8

2pir = pi r^2
2=r
Please note that the value of the area = value of circumference length. But in reality, area cannot = length.

Answer 9

this is the only time they radius will be 2?

Answer 10

Circumference is (2)(pi)(r)

Area is (pi)(r)(r).

If the two are equivalent, then r = 2