# Concentric Circles

davegage
Posers and Puzzles 15 May '05 03:27
1. 15 May '05 03:27
Suppose you have a circle of radius R1. If you draw a chord of fixed length, c, and then slide the chord around the circle (keeping the chord length fixed), the midpoint of the chord will sweep out a concentric circle. Let the area between this concentric circle and the initial circle be A1.

Suppose you have also a different circle of radius R2 > R1. If you repeat the experiment with the same chord of length c, you will create an area A2, which is defined in the same analogous way as A1.

Which is larger -- A1 or A2?
2. 15 May '05 06:56
I am not very familiar with english terms - what is a chord?
3. 15 May '05 07:05
The area between the two concentric circles can be expressed independently of the radii as Ai= Pi.c²/4.
4. 15 May '05 08:55
Originally posted by Mephisto2
The area between the two concentric circles can be expressed independently of the radii as Ai= Pi.c²/4.
Correct. It was sort of a trick question the way I worded it, because it turns out the area is the same regardless of the size of the circle.

BTW, how did you post that superscript 2?
5. 15 May '05 09:04
Originally posted by davegage
Correct. It was sort of a trick question the way I worded it, because it turns out the area is the same regardless of the size of the circle.

BTW, how did you post that superscript 2?
My (dutch/flemish) 'azerty'-keyboard has one key, just above the left/right arrow with two superscripts: x² and x³
6. 15 May '05 19:29
Originally posted by yevgenip
I am not very familiar with english terms - what is a chord?
A chord is just a line which has endpoints that lie on the same circle. So pick any two points on a circle, and draw a line connecting them -- that's a chord.

Note that for any circle, there are infinitely many chords which have length equal to the circle's diameter. If you consider this case, then it is trivial to find the magnitude of the area Ai, and you should quickly get the answer Mephisto posted.