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The display below shows three atoms packed together in a closest-packed arrangement. In the very center of this structure is a hole.
This hole is said to be a **trigonal hole**, because the hole is surrounded by three atoms (the coordination number is three).

Take note that the size of this hole is quite small. If an atom is sufficiently small, it can fit into this hole. Click on the "Show Atom in Hole" to insert a small atom (colored red) in the hole.

**What is the largest atom that can fit into the hole without pushing the outer (blue) atoms apart?**

The radius of the outer atoms is designated *r*. The radius of the hole (this is the radius of the largest sphere that can fit in the hole) is represented by *r _{hole}*.

**What is the ratio r_{hole}/r?**

Derive this ratio using the triangular geometry of the outer (blue) atoms and the laws of trigonometry.

Hint: Begin by drawing an equilateral triangle with sides of length 2*r*. Recall that the sum of the angles around a specific point, such as the
center of the hole site, equals 360^{o}. Finally, remember the definitions of the various trigonmetric functions.

When you have completed your derivation, you may check your answer.

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Holes in Closest-Packed Structures: Trigonal Tetrahedral Octahedral Cubic

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