I am stuck with a question.
Given a circle with radius $x$ cm, what is the maximum number of equilateral triangles of side length 1 cm that can fit in the circle without overlapping or overflowing the circle? $x$ is an integer.
How do I approach this question?
It is easy to give an upper bound for this problem:
The total area of the triangles must not exceed the area of the circle.
\begin{align} n \, A_t & \le A_c \iff \\ n \left( \frac{1}{2} b \, h \right) & \le \pi x^2 \iff \\ n \left( \frac{1}{2} \left( 1 \sqrt{1-\left(\frac{1}{2}\right)^2} \right) \right) & \le \pi x^2 \iff \\ n \frac{\sqrt{3}}{4} & \le \pi x^2 \iff \\ n & \le \frac{4\pi}{\sqrt{3}} x^2 \end{align}
The exact solution I do not know, e.g. see the link given by Gerry Myerson in the comments:
It seems the numbers $n(x)$ might be one of those many hitherto unpredictable, just given and found by trial, discrete properties.
Here is a visualization:
The case $n=6$ has minimal waste of the displayed cases.