Track / Stage 4 / 225 of 340 #485 of 1964

Problem 485

AMC 12 late, AIME early

Geometry Difficulty 4.9 Find the answer

In a unit cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ , if we use the planes $A B_{1} C$ , $B C_{1} D$ , $C D_{1} A$ , $D A_{1} B$ , $A_{1} B C_{1}$ , $B_{1} C D_{1}$ , $C_{1} D A_{1}$ , and $D_{1} A B_{1}$ to cut this unit cube, then the volume of the part containing the center of the cube is $\qquad$ .

Your answer

A number or a short expression. Spacing, $ signs and \frac vs / are all fine.

Official solution

(2) $\frac{1}{6}$ Hint: The part containing the center of the cube is a regular octahedron with vertices at the centers of the cube's faces, and its volume is $\frac{1}{3} \times \frac{1}{2} \times \frac{1}{2} \times 2=\frac{1}{6}$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.