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With equal angles and equal side lengths, what more could you want from a polygon?

The above diagram shows \(3\) identical regular hexagons with side length \(7.\) If \(O_{1}, O_{2}\) and \(O_{3}\) are their respective centers, what is the area of \(\triangle O_{1}O_{2}O_{3}?\)

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The above diagram is part of a regular polygon with \(n\) sides whose center of gravity is \(O.\) If \(n=10\) and the length of a side of the polygon is \(30,\) what is the ratio of the areas of the inscribed circle and circumscribed circle?

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The above diagram is a triangular prism whose faces are all regular polygons. If the length of \(\overline{AB}\) is \( 3,\) what is the volume of the triangular prism?

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The length of each side in the above regular octagon is \(8.\) What is its area?

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