Square Flower
The diagram shows 8 circles surrounding a region. What is the perimeter of the shaded region?
Age
14 to 16
Challenge level
Problem
Image
The shaded region is bounded by eight equal circles. The centres of the circles are at the corners and midpoints of the sides of a square.
The perimeter of the square is 8cm. What is the perimeter of the shaded region?
If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.
Student Solutions
The square has side length $2$cm, so each of the circles has radius $\frac{1}{2}$cm.
The circumference of a single circle, $2\pi r$, is therefore $\pi$cm.
The shaded region is bounded by $4$ halves of a circle and $4$ quarters of a circle.
Therefore the length of the perimeter of the shaded region is $$(4\times \frac{1}{2}+4\times\frac{1}{4})\pi=4\times \frac{3}{4}\times \pi= 3\pi cm.$$