Draw a two perpendicular diameters, dividing the circle into 4 equal areas. In order to fit into one of those quarters, the small circle must have center where the distance from that center to the large circle is equal to the distance from the center perpendicular to one of those diameters. If we let the distance from the center of the large circle to the feet of those perpendiculars be "x", then the distance from the center of the large circle to the center of the small circle is and the distance from the center of the small circle out to the large circle is (R is the radius of the large circle). Then we must have so and . That small circle, then, has radius and so area .There are four equal circles within the circles, taking up the exact same space and such, just at the four quarters of the circle.
If you imagine it, in the middle between these 4 circles is a diamond shaped thingy.
What is the area of the largest circle than can fit in there?
The answer is triple square route of 25pi but how does that work? And why?
Now look at the square formed by those four centers of the small circles. It has sides of length and so area .
The region you want is that square with four quarter small circles taken out. But those four quarter circles together have area equal to the a single small circle. The area of the center region is the area of the square minus the area of the small circle:
I'm not clear on what you mean by "triple square root" but I don't that could be "triple square root of 25 pi".