Inspired by Abhay Kumar

Geometry Level 5

Two quarter circles are inscribed in a square of side length 5. A regular octagon is inscribed in between the two quarter circles and a circle of radius r r is inscribed in between the regular octagon and the two quarter circles. What is r r ?


The answer is 0.33489.

Dexter Woo Teng Koon by Dexter Woo Teng Koon , 20, Malaysia

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2 solutions

Let a be the side of the 8-gon. To simplify calculations, I have introduce a variable X = a ∗ ( 2 + 1 ) X=a*(\sqrt2 + 1)
Solving right triangle GBE will give us the value of X. How to obtain the side lengths is shown in middle top Fig.
The top last Fig. shows GBE with sides in term of X.
On applying Pythagoras Theorem we get a quadratic,
3 4 ∗ X 2 + 5 2 ∗ X − 1 4 ∗ 75 \frac 3 4*X^2 + \frac 5 2*X - \frac 1 4 *75 .
On solving to get a +tive answer we get X = 5 3 ∗ ( 10 − 1 ) X=\frac 5 3*(\sqrt{10} -1 )
After getting the value of X, solving right triangle ODE will give us the value of r. How to obtain the side lengths is shown in bottom two Fig.
The last bottom Fig. gives the right triangle ODE in terms of X, r and constants.
Solving this by Pythagoras Theorem we directly get value of r, since we now know the value of r. the r 2 r^2 term is canceled off.


4 Helpful 0 Interesting 0 Brilliant 0 Confused
Ahmad Saad
May 25, 2016

Let the hexagon have side length a a and height b b .
From regular polygons, we know that b = a ( 2 + 1 ) b = a ( \sqrt{2} + 1 ) .
We can label the other lengths in the diagram:

1 Helpful 0 Interesting 0 Brilliant 0 Confused

That's a very nice short solution!

You can improve the presentation by defining the terms that you're using, so that others can easily understand what you're saying, instead of having to dig that information out of the picture.

Calvin Lin Staff - 5 years ago

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Sir,what terms that required to be explanation. I think that figure attachment is enough.

you can add any information would be improve the solution to others.

Thanks for your comment.

Ahmad Saad - 5 years ago

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My point is that you should make it explicit that a a and b b are the length and height of the hexagon. Because you labelled a lot of other lengths too, it is not immediately apparent what these are supposed to be. I've edited your solution accordingly.

Calvin Lin Staff - 5 years ago

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@Calvin Lin – Thanks. I appreciated your time.

Ahmad Saad - 5 years ago

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