How to find the Relation between Areas Inside a Square

Find the relation between shaded areas inside a square, which is separated by three semicircle

The figure shows a square. Three semicircles are drawn inside the square and divided into six parts. Then find the relation between areas Inside a square (Blue area: Red area)

Solution

We can find the relation between areas using symmetry

From the above figure

shaded areas are equal (red = blue)

So we can interchange these areas then we get a figure like this

Let the sides of the square = 2x, then

From figure

Blue area = ½ πx²

and, Red area = 4x² – ½πx²

So

Blue Area / Red Area = ½πx² / (4x² – ½πx²) 

so, Blue area: Red Area = ½π: 4 – ½ π

Thus, Blue area: Red Area = π: 8 – π

Relation between shaded areas is π: 8 – π