Brain Teasers
Square In Square
A
regular octagon is inscribed inside a square. Another square is inscribed inside
the octagon. What is the ratio of the area of the smaller square to the area of
the larger square?
This brainteaser was
written by Derrick Niederman.
Solution:
.
One way to see this is to rotate the blue triangles onto the
top of the red trapezoids, with the hypotenuse of the triangle flush with the
shorter base of the trapezoid. Then it’s pretty easy to see that if the four
red and blue triangles are folded over, they’d completely cover the yellow
square. In other words, the area of the yellow square is equal to the area of
red trapezoids and blue triangles combined, so the ratio of the smaller square
to the larger square is .
It is also possible to calculate the area of the larger and
smaller squares. Start by assuming that the length of the shorter sides of the
triangles is 1 unit, as shown below. Then the hypotenuse of each triangle
is 
, and since the hypotenuse is also a side of the regular
octagon, then all sides of the octagon are .
Consequently, the side length of the larger square is 
, and the side length of the smaller square is 
, so their respective areas are 
and 
= 
. Dividing the area of the smaller square by that of the
larger yields .
If those calculations are a little too messy for you, then
here is an alternative solution. Note that the length of the diagonal of the
smaller square equals the distance between opposite sides of the regular
octagon. However, the side length of the larger square is also equal to the
distance between opposite sides of the octagon. Therefore, the side length of
the larger square is times the side length
of the smaller square, so the ratio of the areas is .
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