MathDB

Three 12 cm

\times

12 cm squares are each cut into two pieces

A

and

B

, as shown in the first figure below, by joining the midpoints of two adjacent sides. These six pieces are then attached to a regular hexagon, as shown in the second figure, so as to fold into a polyhedron. What is the volume (in

\text{cm}^3

) of this polyhedron?
[asy] defaultpen(fontsize(10)); size(250); draw(shift(0, sqrt(3)+1)*scale(2)*rotate(45)*polygon(4)); draw(shift(-sqrt(3)*(sqrt(3)+1)/2, -(sqrt(3)+1)/2)*scale(2)*rotate(165)*polygon(4)); draw(shift(sqrt(3)*(sqrt(3)+1)/2, -(sqrt(3)+1)/2)*scale(2)*rotate(285)*polygon(4)); filldraw(scale(2)*polygon(6), white, black); pair X=(2,0)+sqrt(2)*dir(75), Y=(-2,0)+sqrt(2)*dir(105), Z=(2*dir(300))+sqrt(2)*dir(225); pair[] roots={2*dir(0), 2*dir(60), 2*dir(120), 2*dir(180), 2*dir(240), 2*dir(300)}; draw(roots[0]--X--roots[1]); label("

B

", centroid(roots[0],X,roots[1])); draw(roots[2]--Y--roots[3]); label("

B

", centroid(roots[2],Y,roots[3])); draw(roots[4]--Z--roots[5]); label("

B

", centroid(roots[4],Z,roots[5])); label("

A

", (1+sqrt(3))*dir(90)); label("

A

", (1+sqrt(3))*dir(210)); label("

A

", (1+sqrt(3))*dir(330)); draw(shift(-10,0)*scale(2)*polygon(4)); draw((sqrt(2)-10,0)--(-10,sqrt(2))); label("

A

", (-10,0)); label("

B

", centroid((sqrt(2)-10,0),(-10,sqrt(2)),(sqrt(2)-10, sqrt(2))));[/asy]

Problem Statement