Circle perimeters (Posted on 2002-11-19)

Prove that the sum of the perimeters of the smaller circles in the Bigger circle is equal to that of the bigger circle. (The centres are on the line given which is the diameter of the bigger circle.)

	Submitted by Dulanjana
Rating: 2.3846 (13 votes)
Solution:	(Hide)
By considering the formula for a circle's perimeter (circumference), the solution becomes fairly obvious: L = 2πR Label the big circle's radius Rb, and the four smaller radii R1, R2, R3 and R4. It's easy to see that R1 + R2 + R3 + R4 = Rb. The sum of the small circles' perimeters is 2πR1 + 2πR2 + 2πR3 + 2πR4 or 2π * (R1 + R2 + R3 + R4). From the equation above we can substitute in the sum of small radii for the large one to get 2πRb, or the perimeter of the large circle.

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