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* = 2*x* = *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 – 2*x*)/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 + *x*2 = 12

2*x*2 + *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 2*x*, so the area will be the square of that value:

(2*x*)2 = (0.5(√7 – 1))2

(2*x*)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