Problems Plus 15: Packing Infinite Circles Inside an Equilateral Triangle

In this video I determine the limit that the area of an infinite number of circles that can be packed inside an equilateral triangle. I represent this area as a ratio of the area of the circles divided by the area of the triangle. To solve this problem, I first determine an expression for the radius of the circles in terms of the length of the triangle. I then count up the number of circles, and determine the equation of the area of n circles. Then, I plug in the length of the triangle as a function of the triangle's area. This then obtains a division of the area of the circles divided by the area of the triangles. Taking the limit as n, the number of rows of circles, approaches infinity, we then obtain the limit we were asked to find. Truly amazing stuff!

The timestamps of key parts of the video are listed below:

• Problem 15: Packing infinite circles inside an equilateral triangle: 0:00
• Solution: Area of an Equilateral Triangle: 1:24
• Area of the circles: 4:34
• Length in terms of 4 radii: 8:48
• Length in terms of n radii: 10:16
• Solving for radius in terms of the length: 13:06
• Counting the number of circles: 13:44
• Total area of the circles: 15:20
• Limit of area of circles divided by area of triangle: 20:02

This video was taken from my earlier video listed below:

• Infinite Sequences and Series: Problems Plus: https://youtu.be/zjdkQIIdTbg
• HIVE video notes: https://peakd.com/hive-128780/@mes/infinite-sequences-and-series-problems-plus
• Video sections playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FQ96Egr5R7fZGeDTIUKz8P

Related Videos:

Sequences and Series playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0EXHAJ3vRg0T_kKEyPah1Lz .

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