Finding What Percentage a Plane is Covered By Pennies Touching Tangentially

Question

The question is the following:

Imagine covering an unlimited plane surface with a single layer of pennies, arranged so that each penny touches six others tangentially. What percentage of the plane is covered?

I can't seem to visually understand how a penny would touch six other pennies tangentially and I am, therefore, unable to understand how the plane is being covered in that pattern. Any help will be greatly appreciated.

Answer 1

Consider this image:

The outer six pennies are touching the center penny tangentially. If you filled a plane with pennies like this, the shaded rectangular area would repeat.

If you calculate the area of the shaded rectangle, $A_{shaded}$, and the area covered by the pennies within the shaded rectangle, $A_{covered}$, the percentage of a plane covered by pennies touching tangentially is $$100\% \cdot \frac{A_{covered}}{A_{shaded}}$$

Because the scale does not matter, you can assume the radius of the pennies to be $1$. (The scale does not matter, because you can use "the radius of a penny" as your measurement unit. Then, the area units are "the radius of a penny, squared". Because the area units cancel out, you are free to choose your length (and therefore area) units.)

To find out the height of the rectangle, consider the angle the centers of two outer pennies make with respect to the center of the center penny.

Answer 2

Draw a regular hexagon and put a penny in the middle and on each of the vertices. That gives you the covering you want.