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Olimpiadi di Matematica - GARA di SECONDO LIVELLO

Geometry Difficulty 4.3 AIME Find the answer Italy

Problem:

Inside a circle of radius $1$ , $3$ arcs of a circle, also of radius $1$ , are drawn, centered at the vertices of an equilateral triangle inscribed in the circle. What is the area of the shaded region?

Figure for this problem

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Solution

Solution:

The answer is (D). The arcs drawn inside the circle have the same radius as the circle itself. As a consequence, by equidecomposition of the circle, the area of the shaded part in figure 1 is equal to that of the shaded part in figure 2, that is, it is equal to the area of the hexagon inscribed in a circle of unit radius. The side of the hexagon is equal to the radius (that is, it equals $1$ ); the area of the hexagon is $6$ times that of the triangle $OPQ$ .

figura 1

figura 2

As a consequence,
$\mathrm{A}_{\text{esagono}} = 6 \cdot \frac{1}{2} \cdot 1 \cdot \frac{\sqrt{3}}{2} = \frac{3 \sqrt{3}}{2}.$

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Source: MathNet, licensed CC-BY-4.0. Statement translated into English from it; metadata (topic, difficulty) added by this project.