This is a problem that I'm attempting to solve (NOT homework!):

There are $10$ trees. Each one is planted in the center of the circle of space that has a diameter of $16$. The $10$ circles (each with diameter of $16$) are placed in the shape of an equilateral triangle ($4$ circles, then $3$ circles on top, $2$ circles on top, $1$ circle on top) so that each side of the triangle is made up of $4$ circles. The circles are as close together as they can possibly be without overlapping.

I am drawing a triangle around the "triangle of circles". My triangle has sides that are as close to the circles as possible without overlapping them. What is the length of the sides of the outer triangle? What is the area of that triangle?

I would appreciate also if you explain how you came to your answer. Thank you!

Here is a picture of the word problem I am describing:

Picture of word problem


If the radius of circle is $R$, then I get the length of the triangle as $(6+2\sqrt{3})R$.

Drop a perpendicular from the center of the rightmost bottom circle to the bottom side.

The distance from the bottom right vertex of the triangle to the base of the perpendicular is $D = \sqrt{3}\ R$, as we get a triangle of angle $30^{\circ}$ and we get $\tan(30^{\circ}) = \frac{R}{D}$. Since $\tan(30^{\circ}) = \frac{1}{\sqrt{3}}$, we get $D = \sqrt{3}\ R$.

Now drop perpendiculars from the centers of the bottom circles to the bottom side and add up the lengths of the segments formed on the bottom side.

Area of an equilateral triangle of side $a$ is $\sqrt{3}\ a^2/4$.


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