3
$\begingroup$

Inspired by PSE#131293 Two circles and a pentagon and its predecessors, here's a hendecagon puzzle:

blue hendecagon with three vertically aligned circles: a red circle whose diameter is the bottom edge, a green circle which touches the top of the red circle and the top vertex of the hendecagon, and a yellow circle centered within the green circle

The red circle's diameter is the bottom hendecagon side and its radius equals 1. The green circle goes through the opposite hendecagon vertex and touches the red circle. The yellow circle has same center as the green circle and radius 1/3th of the green circle's radius.

Calculations allowed but less calculations preferred

Question: is the yellow circle radius smaller than or equal to or larger than 1?

Improve this question
$\endgroup$
0

1 Answer 1

Reset to default
4
$\begingroup$

Answer:

Smaller.

Explanation:

Consider the figure below. Let BL = 1 be the radius of the red circle. Place three more red circles above the original red circle. Then their height from the base of the hendecagon is 7. On the other hand, the height of the hendecagon is cot(π/22), where angle BGL = π/22. Since tan(x) > x + x3/3 for 0 < x < π/2, we have tan(π/22) > 1/7 (interesting note: π < 22/7). Therefore, cot(π/22) < 7, so the yellow circle is smaller.

hendecagon and circles

Improve this answer
$\endgroup$
1
  • 1
    $\begingroup$ fwiw: I like the, agreed, well known, 'interesting' note: (π < 22/7) because it happens to be handy for PSE#131635 $\endgroup$
    – FirstName LastName
    Commented Jun 12 at 0:23

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.