The shaded figure is formed by eight circular arcs of equal radius. Four arcs are centered at the vertices of a square, and the other four at the midpoints of its sides. The diagonals of the square have length 1.

Find the total length of the boundary of the shaded figure.

As usual, I encourage you to give this problem a go, and when you are reading, keep reading for the solution!

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Solution

Let’s take a look at these two diagrams.

I have colored the one on left red. These are arcs whose centers are the vertices of the square, and they form semi-circular arcs.

On the left, we have green arcs whose centers are the midpoints of
the sides of the square, and they form three-quarter circles.

If each of these arcs has radius r, then the length of the border is

Let’s pay attention to the line PQ joining he midpoints of two adjacent sides of the square.

The line spans two radii of the. And we can see that the diagonal of the square span 4 radii.

Because we are given the diagonal has length 1, the line PQ must have length 1/2.

If 2 radii make up the length 1/2, that means the radius of these arcs is 1/4.

So the the length of the border of the shaded design is