# 2005 PMWC Problems/Problem I4

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## Problem

The larger circle has radius 12 cm. Each of the six identical smaller circles touches its two neighbours and the larger circle. What is the radius of the smaller circle?

$[asy] unitsize(0.5cm); draw((0,3)..(3,0)..(0,-3)..(-3,0)..cycle); for (int i=0;i<6;i=i+1){ draw(dir(60*i)..3*dir(60*i)..cycle); } [/asy]$

## Solution

$[asy] unitsize(0.5cm); draw((0,3)..(3,0)..(0,-3)..(-3,0)..cycle); for (int i=0;i<6;i=i+1){ draw(dir(60*i)..3*dir(60*i)..cycle); } draw((0,1)..(0,-1)..cycle,rgb(1,0,0)); [/asy]$ Adding a circle in the middle makes the solution obvious. Since the radius of the big circle is $12$ cm, then the diameter of the big circle is $24$ cm. But this is also $3$ of the little circle's diameter! Therefore, the diameter of the little circle is $24\div3=8$ cm and the radius is therefore $\boxed{4 \text{cm}}$.

## See also

 2005 PMWC (Problems) Preceded byProblem I3 Followed byProblem I5 I: 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 T: 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10