Question number 12. The square shown in the picture to the right over here has a side length of 2. The four shaded circles are centered at the midpoints of the sides of the square and are tangent to the larger arcs. The larger arcs are centered at the vertices of the square and they intersect here at its center. The question is what is the sum of the areas of the shaded circles? So to find the areas we have to keep track of the radii of the circles so I will draw in a radius here of the blue circle like so for example and I will call that radius maybe let me draw in another one let me call that the radius lowercase r and then I can also draw in a larger radius here in red for the large arc here like that and let me call that capital R and if I extend the radius all the way across since these radii make up half the diagonal we have the diagonal of the square here like that and that's because the arcs intersect at the center so that distance over here the diagonal let me label that as 2 times capital R and then what we have is finally a side length of of 2 over here for the square so using the Pythagorean theorem we can compute the diagonal 2r and so what we have is the hypotenuse squared quantity 2 capital R squared is equal to 2 squared plus 2 squared so 4r squared is equal to 8r squared is equal to 2 and so we know that the large radius is the square root of 2 and we can use that to compute the smaller radius so what we have is let me draw in one more red radius here that's that's the capital R and here we have a smaller distance let me call that X and so we see that X plus capital R is exactly a side length of the square so that's 2 and so solving that for X we have X is equal to 2 minus capital R so 2 minus the square root of 2 and what do we have then we have that the diameter of the blue circle that's 2r and that is equal to the side length 2 minus 2 X and that allows us to solve for little r so r is equal to 1 minus X and that is 1 minus and what we have then is 2 minus the square root of 2 from our previous calculation so lowercase r is equal to square root of 2 minus 1 and so the area of one shaded circle then comes out to we know the formula pi r squared so pi and then copy the radius which is root 2 minus 1 quantity squared and so that's for one of the circles there are four of them so we just multiply that answer by 4 and it looks like choice A so we have found our answer here four times pi times the radius quantity root 2 minus 1 squared is choice A

square

circles

Pythagorean theorem

radius

area