# Chapter 9 - Section 9.1 - Prisms, Area, and Volume - Exercises - Page 409: 49

The length of the diagonal is $\sqrt{3}~e$

#### Work Step by Step

The edge of the cube has a length of $e$ Let vertex A be a vertex on the bottom of the cube. Let vertex B be the vertex on the bottom of the cube that is diagonally opposite vertex A. We can calculate the the length $d$ of a diagonal across the bottom surface of the cube joining vertex A and vertex B: $d = \sqrt{e^2+e^2} = \sqrt{2}~e$ Let vertex C be a vertex at the top corner of the cube adjacent to vertex B. The three vertices A,B, and C form a triangle. We need to calculate the length $L$ of the line $\overline{AC}$: $L = \sqrt{(\overline{AB})^2+(\overline{BC})^2}$ $L = \sqrt{(\sqrt{2}~e)^2+(e)^2}$ $L = \sqrt{2~e^2+e^2}$ $L = \sqrt{3~e^2}$ $L = \sqrt{3}~e$ The length of the diagonal is $\sqrt{3}~e$

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