It's a kinda basic geometry calculation. The link flixen provided :
http://mathcentral.uregina.ca/QQ/database/QQ.09.08/h/anthony1.html
is the easiest way to calculate this matter. I also did this problem with this solution
also rechecked with the "more complex" calculus
154422 is the best answer.
I will try my best to explain how to calculate this :
1st : just denote the side of the square by A, or just 1cm (simpler calculations).
2nd : as you can read Anthony's solutions, we have to find the area of 1/4 of the area, and at first we have to find the contact points of the quarter-circles. Denote the position of P as (0,0) (Cartesian coordinates) > easy to find out the contact points have the position (1/2, (sqrt3-1)/2) and ((sqrt3-1)/2,1/2).
3rd : with the contact point now you may use calculus to solve this problem, but it's complex
so just forget it and focus on Anthony's solution.
As you can see in Anthony's solution, we can divide the 1/12 circle (with the contact points' position it's easy to show the angle of the area is 30 degree) contains the 1/4 area of region R we gonna calculate into 3 regions : 2 equal triangles and the 1/4 area of R.
4th : calculate the area of the triangles : the angles of each triangle : 15, 135 and 30 degree, the sides : 1/ sqrt2, 1/4 and (sqrt3 - 1)/2. So you can easily calculate the area of them with the equation : A = a*b*sinC / 2 (the area of a triangle ABC with sides of a,b,c) .
Or you may use Heron's formula :
http://en.wikipedia.org/wiki/Heron%27s_formula
5th : The last part : Calculate the 1/12 "cake" area then minus the area of 2 triangles, and multiply by 4, and then multiply with the square of the side of the outer square (700cm*700cm) so we can receive the answer :
(pi/12 - (the area of one triangle)*2 ) * 4 * 700^2.
And that's all
hope you can follow the instructions and solve this on your own (even not perfectly on your own
).