If the area of a circle increases by a factor of 9, by what factor does its circumference increase ?

a = pi r^2

A = pi R^2 = 9 a = 9 pi r^2
so
R^2 = 9 r^2
R = 3 r

c = 2 pi r
C = 2 pi R = 2 pi (3r) = 3 (2 pi r)
so
C = 3 c

Since A = pi*r^2

9A = pi*(3r)^2
That is, the radius grows by a factor of 3

So, since C = 2pi*r, it also grows by a factor of 3.

Just so you know, if
the linear value grows by n,
the area grows by n^2
the volume grows by n^3

C= 2 pi r = 2 pi (3r) = 3 ( 2 pi r )

so C = 3 c

why C = 3 c I don't understand

Well, you know what they say, "circumferences have a lot of layers." If the area of a circle increases by a factor of 9, it means that the radius increases by a factor of √9, which is 3. And since the circumference of a circle is calculated using the formula 2πr, when the radius increases by a factor of 3, the circumference would increase by the same factor. So, the circumference would increase by a factor of 3, just like that!

To determine the factor by which the circumference of a circle increases when the area increases by a factor of 9, we need to understand the relationship between the two.

Let's start with the formulas for the area and circumference of a circle:

Area of a circle = π * r²
Circumference of a circle = 2 * π * r

In these formulas, "r" represents the radius of the circle.

Now, let's consider the relationship between the area and radius of a circle. The area of a circle increases with the square of the radius. This means that if the radius is multiplied by a factor of "x," then the area will increase by a factor of "x^2."

Now, let's apply this understanding to the problem. If the area of a circle increases by a factor of 9, it means that the radius is multiplied by the square root of 9, which is 3. So, the radius is multiplied by a factor of 3.

Since the circumference of a circle is directly proportional to the radius, the circumference will also increase by the same factor. Therefore, the factor by which the circumference increases is 3.