Crazy shading

Can you work out the fraction of the larger square that is covered by the shaded area?
Age
14 to 16
Challenge level
filled star empty star empty star
Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving
Being curious Being resourceful Being resilient Being collaborative

Problem

Crazy Shading



The diagram show four circles each of which touches the largest square and two adjacent circles. A second square has its vertices at the midpoints of the sides of the largest square and the central square has its vertices at the centres of the circles.

 

What is the ratio of the total shaded area to the area of the outer square?

If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.