A geometry problem by Yuan Zhi Lee

Geometry Level 1

There are n number of circles inscribed in a larger circle. The diameter of the smaller circles fully superimpose that of the larger circle( they are in alignment with the diameter of the larger circle),and the sum of the diameters of the smaller circles is equal to that of the larger circle. If the circumference of the larger circle is 78.9 cm , what is the sum of the circumferences of all the smaller circles?

The answer is 78.9.

1 solution

Yuan Zhi Lee
Mar 10, 2018

Since the diameter of the smaller circles add up to the larger circle, and the fact that the diameter of the smaller circle is in alignment with the larger circle:

We would know that: $D_Larger circle=D_1+D_2+...+D_n$

From the circumference formula: $2 πR=78.9cm^2$ And the fact that $2R=D_Larger circle=D_1+D_2+...+D_n$

We can substitute $2R$ for $(D_1+D_2+...+D_n)$ , yielding the equation $(D_1+D_2+...+D_n)π = 78.9cm^2$

From here, it should be apparent to you that $D_n=2r_n$

Further subsituting this into $(D_1+D_2+...+D_n)π = 78.9cm^2$ yields $(2r_1+2r_2+...+2r_n)π$ which leads to one realizing that the answer is $78.9cm^2$

A geometry problem by Yuan Zhi Lee

The answer is 78.9.

1 solution

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