[[!redirects four equilateral trangles round a square solution]] # Four Equilateral Triangles Round a Square +-- {.image} [[FourEquilateralTranglesRoundaSquare.png:pic]] > Four equilateral triangles are arranged around a square which has area $12$. What's the total shaded area? =-- ## Solution by [[equilateral triangle|Properties of an Equilateral Triangle]] Let $x$ be the side length of the inner square, so that $x^2 = 12$. From the relationships between the [[lengths in an equilateral triangle]], the height of the equilateral triangles is $\frac{\sqrt{3}}{2}$ and so the diagonal of the outer square is $x + \sqrt{3}x$. The area of the outer square is therefore: $$ \frac{1}{2} (x + \sqrt{3} x)^2 = (2 + \sqrt{3}) x^2 = 24 + 12\sqrt{3} $$ The area of each equilateral triangle is $\frac{\sqrt{3}}{4} x^2 = 3 \sqrt{3}$ and so the area of all four is $12 \sqrt{3}$. Hence the shaded area is $24 + 12 \sqrt{3} - 12 - 12 \sqrt{3} = 12$.