Question 833336
{{{drawing(300,300,-6.5,6.5,-6.5,6.5,
red(circle(0,0,6)),
circle(0,0,2),circle(4,0,2),circle(-4,0,2),
circle(2,3.464,2),circle(-2,3.464,2),
circle(2,-3.464,2),circle(-2,-3.464,2),
locate(2.4,0.4,6in),locate(-1.6,0,2in),
green(arrow(-2,0,0,0,0)),green(arrow(0,0,-2,0)),
green(arrow(2.3,0,0,0)),green(arrow(3.7,0,6,0))
)}}} {{{highlight(7)}}}cookies
The least unused space between cookies and the edge of the dough circle will be achieved when the cookies are against the edge of the dough circle.
The least unused space between cookies will be achieved when the cookie centers are forming an equilateral triangle. Three {{{1/6}}} of a cookie wedges will be inside that triangle. The area of that {{{3/6=1/2}}} of a circle of radius {{{R}}} is {{{pi*R^2/2}}} , while the triangle, with sides measuring {{{2R}}} will have a height of {{{sqrt(3)*R}}} and an area of {{{sqrt(3)*R^2}}} .
The fraction of that area covered by the circle wedges is
{{{(pi*R^2/2)/(sqrt(3)*R^2)=pi/2sqrt(3)}}}= approx. 0.91 (91%).
If you place circles with radius {{{R}}} so that the centers form a square, the side of that square will be {{{2R}}} and the area of that square will be {{{(2R)^2=4R^2}}} , while the 4 quarter circle wedges covering that square will have a total area of {{{pi*R^2}}} , amounting to
{{{pi*R^2/4R^2=pi/4}}}= approx. 0.79 (79%) of the square area.