# concentric circles / chord & tangents

#### wyworocksx89

##### New member
I have this problem that I can't figure out, and it's driving me nuts. I don't know if I'm just being stupid or what...okay, well it's a diagram, so I'll try my best to describe it:

Two circles are concentric (they have the same center). There is a chord of the bigger, outer circle which is tangent to the smaller, inner circle. This chord's total length is 20. Find the radius of the smaller circle.

Thanks to anyone who can help!!
-kiki

#### pka

##### Elite Member
In order for you to have a unique solution there must be more information!
If r is the radius of the inner circle and R is the radius of the outer circle then:
$$\displaystyle r = \sqrt {R^2 - 100}$$.

#### soroban

##### Elite Member
Hello, kiki!

Is that the original wording?
It sounds suspiciously like a classic problem.

Two circles are concentric (they have the same center).
There is a chord of the outer circle which is tangent to the inner circle.
This chord's total length is 20. Find the radius of the smaller circle.
The classic problem has the punchline:
$$\displaystyle \;\;$$Find the area of the annulus (ring) between the circles.

Surprisingly, this problem has a unique answer.

#### wyworocksx89

##### New member
Wait, you're right, that is the problem! I just got to the point where I assumed you needed the radius.

#### pka

##### Elite Member
wyworocksx89 said:
Wait, you're right, that is the problem! I just got to the point where I assumed you needed the radius.
Well then you can use my post!
Area<SUB>O</SUB>−Area<SUB>i</SUB> can be found!

#### wyworocksx89

##### New member
I'm not sure I understand...I don't know r or R (and can't figure out any way to get them), so how would I use that formula?

#### pka

##### Elite Member
The point is that it does not matter!
$$\displaystyle Area_O = \pi R^2$$
$$\displaystyle Area_i = \pi r^2$$.
$$\displaystyle Area_O - Area_i = \pi R^2 - \pi r^2 = \pi R^2 - \pi \left( {\sqrt {R^2 - 100} } \right)^2$$