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Geometry

Regular Polygons

Concept Quizzes

Regular Polygons Warmup
Regular Polygons - Angles
Regular Polygons - Area
Regular Polygons - Properties of Regular Polygons
Regular Polygons - Properties of Regular Hexagons
Regular Polygons - Problem Solving
Regular Polygons - Decomposition into Triangles

Challenge Quizzes

Regular Polygons: Level 1 Challenges
Regular Polygons: Level 3 Challenges
Regular Polygons: Level 4 Challenges

Regular Polygons - Properties of Regular Polygons

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 $\triangle O_{1}O_{2}O_{3}?$

\frac{145}{4}\sqrt{3}

\frac{155}{4}\sqrt{3}

\frac{147}{4}\sqrt{3}

\frac{153}{4}\sqrt{3}

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by Brilliant Staff

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?

\sec^2 72^{\circ}:1

\sin^2 72^{\circ}:\cos^2 72^{\circ}

\sin^2 72^{\circ}:1

\cos^2 144^{\circ}:\sin^2 144^{\circ}

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by Brilliant Staff

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?