Tessellated Arcs

The tessellated figure below consists of 3 equal arcs in an equilateral triangle. Each side measures 2 inches. What is the area of the region labeled X?
Source: Julia Robinson Mathematics Festival


Draw altitude \overline{AH} to side \overline{BC}. Because \triangle {AHC} is a 30^\circ-60^\circ-90^\circ triangle with a hypotenuse of length 2 inches, AH=1\sqrt 3.

Area of \triangle {ABC}=\frac{1}{2}\left (BC\right )\left (AH\right )
=\frac{1}{2}\left (2\right )\left (\sqrt 3\right )
=\sqrt 3

Each arc lies on a circle of radius 1 inch centered at one of the three vertices of \triangle {ABC}. Each arc forms a sector with a central angle measuring 60^\circ. The three sectors if placed contiguously next to each other would therefore form a semicircle.

Area of the three sectors = \frac{180^\circ}{360^\circ}\pi \left (1^2\right )=\frac{\pi}{2}

Area of region X= area of \triangle{ABC}- area of the three sectors
=\sqrt 3-\frac{\pi}{2}

Answer: \sqrt 3-\frac{\pi}{2} square inches.


About mvtrinh

Retired high school math teacher.
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