What's the area? (Posted on 2002-05-08)

Consider the picture to the right. We have two concentric circles, one inside another. We do not know the radius for either one of them.

We do, however know that the chord (shown in red) - a line whose ends are on the outer circle and which is tangent to the inner circle - has a length of 10 inches.

What is the area (shaded in light-blue) between the two circles?

	Submitted by levik
Rating: 3.7333 (15 votes)
Solution:	(Hide)
Ler's say the small circle has the radius r and the big one's is R. Then what we need to find is: (pi)R^2 - (pi)r^2 Let's call the point where the red line touches the inner circle "P", the center of the circle "C" and the red line's endpoints "A" and "B". C-P, is be the radius of the small circle, C-A and C-B - the radii of the large one, and A-B is known to be 10 inches long. Because A-B is a tangent to the inner circle, it is perpendicular to C-P, and A-P-C makes a right triangle, with R for the hypotenuse, and r and 100/2 for legs. Use the pythagorean theorem: r^2 + (5)^2 = R^2 or 25 = R^2 - r^2 Multiply both sides by pi and we get 25(pi) = (pi)R^2 - (pi)r^2 Which was what we were looking for. So the area is 25(pi)

	Subject	Author	Date
	Puzzle Thoughts	K Sengupta	2024-02-20 08:33:28
	Thank you, Nikki	CeeAnne	2004-10-05 14:17:53
	re: Why not post the best solution?	nikki	2004-10-05 07:44:36
	Why not post the best solution?	CeeAnne	2004-10-04 21:24:40
	Solution	Antonio	2003-08-22 05:31:02
	corrected solution	Ravi Raja	2003-03-03 17:34:16
	What's the Area ?	Ravi Raja	2003-01-06 02:37:04
	i don't know you...here is my solution	Daniel	2002-10-15 21:14:43
	Area	TomM	2002-05-09 23:34:50
	Better Answer...	Tom	2002-05-09 16:43:23
	My Answer	Tom	2002-05-09 12:37:27
	I have the solution	Sergio Pasini	2002-05-09 06:59:37
	solution	narcoleptic	2002-05-08 14:34:24
	solution	tony danza	2002-05-08 14:17:34
	More info on Solution	Andrew Elliott	2002-05-08 12:28:47
	Solution??	Fizzle	2002-05-08 12:00:27