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Construct a circle of radius 1. From a point on the circle, construct another circle of radius 1. Inscribe a square where the two circles overlap. What is the area of the square?
I thank Nibedan Mukherjee who sent me the problem and its solution in October 2018. I was also told this problem appeared in the 2019 Senior Mathematical Challenge by the UKMT.
Watch the video for a solution.
What Is The Area? Square Between Two Circles
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Answer To Square Inside Overlapping Circles
(Pretty much all posts are transcribed quickly after I make the videos for them–please let me know if there are any typos/errors and I will correct them, thanks).
Suppose the circles have centers O and P, and label the square as ABCD.
From the center of one circle (with center O), draw a radius to the opposite corner of the square (point C), and draw another line segment to the opposite side of the square (point E).
Because OC is a radius of circle O, we have:
OC = 1
Suppose CE = x. Then by symmetry EB = x. Thus, the side of the square has length given by:
CB = 2x = AB
Since OP is a radius of circle O, its length is 1. The distance from O to AD is the same as P to CB by symmetry (and the latter distance is EP). Therefore, difference of OP and AB gives double the distance of EP. Thus we can then calculate OE as follows:
OP = OE + EP
OP = OE + (OP – AB)/2
1 = OE + (1 – 2x)/2
OE = 0.5 + x
We now consider the right triangle OEC. It has legs of OE = 0.5 + x and CE = x and a hypotenuse of 1. Thus we have:
(0.5 + x)2 + x2 = 12
2x2 + x – 0.75 = 0
We can solve the above equation using the quadratic formula. Since x is a length, it should be positive and we can reject the negative solution. Thus we get:
x = 0.25(√7 – 1)
A side of the square is 2x, so the area will be the square of that value:
(2x)2 = (0.5(√7 – 1))2
(2x)2 = 0.5(4 – √7) ≈ 0.677
References
2019 Senior Mathematical Challenge paper (see question 25)
https://www.ukmt.org.uk/sites/default/files/ukmt/senior-mathematical-challenge/SMC_2019_Paper.pdf
Solutions (see question 25)
https://www.ukmt.org.uk/sites/default/files/ukmt/senior-mathematical-challenge/SMC_2019_Extended_Solutions.pdf
Senior Mathematical Challenge archive papers
https://www.ukmt.org.uk/competitions/solo/senior-mathematical-challenge/archive
