A square in circles

I have come across this problem which I think provides a great problem solving challenge for students.

Four identical circles are arranged in a square as shown in the diagram. If the length of the side of the larger square 4 units long, what is the size of the area of the smaller square?

Have a go before reading on.

Square in circles

At first I did question whether the square in the centre was actually a square. An optical illusion, or my poor eyesight, made me think the edges were bowed in some way. I set myself the challenge of constructing the diagram using Geogebra in such a way that I could use a slider to make the size of the square get bigger and smaller. I could then use Geogebra to check my answer. Have a go, it is a good challenge to set. I have attached my effort.

So to solving the problem. Well I did it essentially using the same strategy, but in two slightly different ways, one using the length of the diagonal of the square and one creating a right angled triangle from the centres of the circles.

I then considered how I can ask students to extend the problem. Write an expression for the area of the small square when the

  • radius of the circle is r
  • length of the square is s
  • the circumference of the circle is p
  • the area of the large square is A
  • the area of the circle is C

I have attached a blank copy of the problem and a copy of my first method. I am interested to find whether students find other, more inventive ways of finding the area. Reply to this post if you receive any



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Note: 'circle' should be 'square' at end of question.

Symmetry can also be used to get:

4 root 2 - 4 = 4 x ,                   where small  square is 2x by 2x.

Stephen Lyon

Many thanks, corrected now