What is the equation for this basic geometry-related statement?

Question

In a zoology journal I read recently, I came across this statement: "The area of the outer circle was 3 times larger than that of the inner circle."

It was also given that the radius of the outer circle was 100 m while the radius of the inner circle was 50 m.

I tried verifying this using the equation: Ao/Ai=3 (Ao=area of outer circle, Ai = area of inner circle). But instead of getting the expected answer of 3, I got 4 instead. I suspect that there was something in the statement that made things quite confusing.

So I ask you, what did the statement mean? And what was the equation that was used for the said statement? I hope you can reply ASAP.

Answer 1

If the radius of the inner circle is 50m then
A=πr²
=2500π
The outer circle would be
=10000π

If by the area of the outer circle they meant the area between the outer and inner then
A = 10000π-2500π=7500π
Then the ratios would be 7500π/2500π=3

Answer 2

I guess the "outer circle" meant the area of the outer circle outside the inner circle (i.e., not counting the inner circle).

Take a 50 m circle. Paint the inside red.
Take a 100 m concentric circle (i.e., same centre). Paint the inside green EXCEPT where it is already red.

Then the green area will be 3 times the red area.

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If that is not it, they it could be a typo.

Answer 3

Ok, this is what I found.

If you have two circles like you described, the outer circle's area will be 4 times the inner circle's area.

BUT

If you think of 2 circles like this, there is the donut area. In other words, the area outside of the small circle but still inside the big circle. It is like a donut shape.

This area will be (10000 - 2500) square meters or 7500 square meters. This donut area is 3 times the area of the smaller circle.

Answer 4

You are correct. The area of the inner circle is pi*(50^2). The area of the outer circle is pi*(100^2). The ratio of areas (pi*(100^2))/(pi*(50^2)) will reduce to 4.