Four Leaf Clover
The diagram shows four equal discs and a square. What is the perimeter of the figure?
Age
14 to 16
Challenge level
Problem
Image
The diagram shows four equal discs and a square. Each disc touches its two neighbouring discs. Each corner of the square is positioned at the centre of a disc. The side length of the square is $\frac{2}{\pi}$. What is the length of the perimeter of the figure?
If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.
Student Solutions
The radius of each disc in the figure is equal to half of the side-length of the square, i.e. $1/\pi$.
Image
Because the corners of a square are right-angled, the square hides exactly one quarter of each disc. So
three-quarters of the perimeter of each disc lies on the perimeter of the figure. Therefore the length of the perimeter is $$4\times\frac{3}{4}\times 2\pi\times\frac{1}{\pi}=6\;.$$