Calculating Distance Between Ions in Sodium Chloride Crystal Structure

[1irishman]

Homework Statement

A drawing shows sodium and chloride ions positioned at the corners of a cube that is part of the crystal structure of sodium chloride. The edge of the cube is 0.281nm in length. Find the distance between the sodium ion located at one corner of the cube and the chloride ion located on the diagonal at the opposite corner.

Homework Equations

I'm thinking pythagorean theorem and one or more of the trig functions perhaps?

The Attempt at a Solution

Well...I figured the diameter if the cube is drawn inside a circle is 2(0.281)=0.562nm.
Not sure what to do this...it's difficult for me to conceptualize.

(The answer in the book is 0.487nm)

 

[HallsofIvy]

Just Pythagoras. If each edge of the cube has length x, then A diagonal of one side, say, the base, is given by $d^2= x^2+ x^2= 2x^2$ so that $d= x\sqrt{2}$. Now, that "face diagonal" together with a vertical edge gives you another right triangle having the diagonal of the cube from one corner to the opposite corner as hypotenuse. $D^2= x^2+ d^2= x^2+ 2x^2= 3x^2$. That diagonal has length $D= x\sqrt{3}$.

 

[1irishman]

so is this two triangles?

 

[1irishman]

maybe I'm confused because of the variables you are using...i'm use to seeing a,b,c with pythagoras equations..i know it doesn't make a difference..but for us newbies it kinda does sometimes make it easier to picture. I"m trying to visualize what this would all look like if flatttened out i guess.