Hello, Vicky!

On each side of a unit square, an equilateral triangle of side length 1 is constructed.

On each new side of each equilateral triangle, another equilateral triangle of side length 1 is contructed.

The interiors of the square and the 12 triangles have no points in common.

Let R be the region formed by the union of the square and all the trianlges,

and let S be the smallest convex polygon that contains R.

What is the area of the region that is inside S but outside R? Code:

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|:::::*-------o:::::|
|::*::| A|::*::|
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The region $\displaystyle R$ is comprised of a unit square and 12 equilateral triangles.

The area of $\displaystyle R$ is: .$\displaystyle A_R \;=\;1 + 12\left(\tfrac{\sqrt{3}}{4}\right) \;=\;1 + 3\sqrt{3}$

$\displaystyle \Delta ABC$ has two sides $\displaystyle AB = AC = 1$ and included angle $\displaystyle \angle BAC - 30^o$

Its area is: .$\displaystyle \tfrac{1}{2}(1^2)\sin30^o \:=\:\tfrac{1}{4}$

The area of $\displaystyle S$ is the area of $\displaystyle R$ plus four of those triangles.

. . $\displaystyle A_S \;=\;(1 + 3\sqrt{3}) + 4\left(\tfrac{1}{4}\right) \:=\:2 + 3\sqrt{3}$

The area inside $\displaystyle S$ and outside $\displaystyle R$ is: .$\displaystyle (2+3\sqrt{3}) - (1 + 3\sqrt{3}) \;=\;1$