A regular dodecagon inscribed in a circle of radius 1 has area 3. (show me this)

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We have a regular dodecagon inscribed in a circle of radius 1. To find its area, we'll use the polygon area formula. A dodecagon has 12 sides, and we can see the central angle between adjacent vertices. Now let's substitute the values into our formula. For a dodecagon, n equals 12, and the radius r equals 1. This simplifies our expression to 6 times sine of pi over 6. We can see one of the 12 triangular sectors highlighted. To complete our calculation, we need to evaluate sine of pi over 6. Pi over 6 radians equals 30 degrees. From the unit circle, we can see that sine of 30 degrees equals one half. Therefore, our area becomes 6 times one half, which equals 3. Here's an alternative geometric approach. We can divide the dodecagon into 12 congruent isosceles triangles. Each triangle has two sides equal to the radius and a central angle of 30 degrees. The area of one triangle is one quarter, so the total area is 12 times one quarter, which equals 3. In conclusion, we have verified using two different methods that a regular dodecagon inscribed in a circle of radius 1 has an area of exactly 3. Both the polygon area formula and the triangle decomposition approach give us the same result, confirming our calculation.