Answer: Factor of 2
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Step-by-step explanation:
Let's say we had a circle with radius 5
It's area would be
A = pi*r^2
A = pi*5^2
A = 25pi
Now let's increase this by a factor of 4. We'll multiply the area by 4
new area = 4*(old area) = 4*(25pi) = 100pi
Now, we'll use this area value to find what the radius must be
A = pi*r^2
100pi = pi*r^2
100 = r^2
r^2 = 100
r = sqrt(100)
r = 10
The old radius r = 5 jumps to r = 10 so that the area quadruples.
We see that the radius has doubled. This must also mean the perimeter around the circle, aka circumference, must also have doubled.
C = 2*pi*r = 2*pi*5 = 10pi = old circumference
C = 2*pi*r = 2*pi*10 = 20pi = new circumference
This is directly because r has been doubled
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Here's one way to think of why this works:
Consider a square of side lengths 3. It's area is 3*3 = 9.
If we double the sides, then we have a 6 by 6 square of area 6*6 = 36.
Note the jump from 9 to 36 is "times 4". This multiplier 4 is from each side doubling and we have 2*2 = 4.
If you were to triple each side, then the area multiplier would be 3*3 = 9
Multiplying each side by k leads to an area multiplier of k^2.