TITLE: What is the relationship between blue area and red area?
QUESTION [11 upvotes]: This is a grade school problem!
Consider the following figure:

It is very easy to show that the red area and the blue area equal. I can demonstrate this based on my knowledge related to the computation of the surface areas of circular sectors and triangles. Both areas equal $2\left(\frac{r^2\pi}4-\frac{r^2}2\right)$ where $r$ is the radius of the smaller circles.


But, how am I going to show the same if I forget, for good, the formula providing the area of a triangle?

I am not able to get rid of my thought process using triangles.

REPLY [2 votes]: I assume the sides of the square are $2R$, and that the blue area is the area of the intersection of two half circles of radius $R$
$RedArea = Sa_{Big_{quartercircle}} - 2 * Sa_{Small_{halfcircles}} + intersection $
But 
$Sa_{Big_{quartercircle}} = \pi R^2$
and:
$Sa_{Small_{halfcircle}} = \pi R^2\div 2$
So:
$RedArea = intersection = BlueArea$