figure it out

My answer is below, between the brackets. Highlight to see it.

[We're given a large circle in which are inscribed four smaller circles, all of the same size and all tangent to each other and to the larger circle. Center points for the larger circle and the smaller circles are all marked. The smaller circles are colored blue; the remainder of the larger circle's interior is colored yellow. We're given that the total area of all four smaller circles is 100π, and our mission is to find the yellow area.

Obviously, the yellow area = Area_{large circle} - Area_{4 smaller circles}. So we first need to find the radii of the smaller circles. Easy enough:

If the total area of the smaller circles is 100π, then the area of a single small circle is 25π. If the formula for a circle's area is πr², then r = 5. If I draw a line segment radiating from the center of the "northwest" small circle to the point of tangency with the large circle, I know that that length is 5. If I do the same with the "southeast" small circle, I get another 5. We're getting closer to finding out the diameter of the larger circle. To find the diameter, I need to know the length of the segment connecting the northwest and southeast small circles' centers.

To figure that out, we must first realize that the centers of the small circles, if all connected by line segments, form the corners of a square. Each side of the square has the length of two small-circle radii, i.e., 5 × 2 = 10. We remember from algebra and geometry that the length of a square's diagonal is √2  times longer than the length of the square's side. So the length of the square's diagonal, in this case, is 10√2.

Knowing this, we add the lengths of the segments we'd calculated two paragraphs above. So the total diameter of the large circle is 10 + 10√2. To get the area of the large circle, divide by 2 to get the radius: 5 + 5√2. The area of the large circle is πr², which means 

(5 + 5√2)²π = 

(25 + 25√2 + 25√2 + 50)π =

75π + 50π√2 

To get the final answer, we subtract the total area of the small circles from the above.

(75π + 50π√2) - 100π =

50π√2 - 25π. In decimals, this is approximately 143.57 units.

QED.]