# Triangles packed into a unit circle

2014 triangles have non-overlapping interiors contained in a unit circle.What is the largest possible value of the sum of their areas?

What are some ideas that might help me start this?

Note that the 2014 points do not have to lie on the circle.

## Answer

The first triangle placed must have 3 edges. A triangle placed after that with the largest area should share a side with the original triangle, and cover 1 original side, and add 2 new sides, giving it an extra side overall. Therefore, with 2013 more triangles than the first one, there should be a 2016 (3+2013) sided figure created for largest area covered. For this 2016-gon to have the largest area, it should be a regular 2016-gon, which can be done with 2014 triangles. The area for this is $\frac{1}{2}nR^2\sin{\frac{360^\circ}{n}}$ with R being the circle radius, and n being the number of sides. The rest can easily be plugged in to find the total area of the 2014 triangles.

Attribution
Source : Link , Question Author : Ayesha , Answer Author : Asimov