Competition problem — Geometry — Maths Olympiad Prep

Library / /110 of 377

Geometry Difficulty 4.9 AIME Prove it United States

Problem:

The corner of a unit cube is chopped off such that the cut runs through the three vertices adjacent to the vertex of the chosen corner. What is the height of the cube when the freshly-cut face is placed on a table?

Solution

Solution:

The major diagonal has a length of $\sqrt{3}$ . The volume of the pyramid is $1/6$ , and so its height $h$ satisfies $\frac{1}{3} \cdot h \cdot \frac{\sqrt{3}}{4}(\sqrt{2})^{2} = 1/6$ since the freshly cut face is an equilateral triangle of side length $\sqrt{2}$ . Thus $h = \sqrt{3}/3$ , and the answer follows.

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.