Math 2.0

Geometry Definition

Regular polygon

Builds on Euclidean planeEuclidean distanceAngle at a pointConcyclic points

Definition

Let n≥3n \geq 3, and let P1P2⋯PnP_1 P_2 \cdots P_n be a convex polygon in the Euclidean plane ℝ2\mathbb{R}^2. Write P0=PnP_0 = P_n and Pn+1=P1P_{n+1} = P_1. The polygon is regular if all sides have the same length, d(Pi,Pi+1)=d(P1,P2)d(P_i, P_{i+1}) = d(P_1, P_2), and all angles are equal, ∠Pi−1PiPi+1=∠PnP1P2\angle P_{i-1} P_i P_{i+1} = \angle P_n P_1 P_2, for i=1,…,ni = 1, \dots, n.

Remarks

Each interior angle of a regular convex nn-gon is (n−2)πn\frac{(n - 2)\pi}{n}. Each exterior angle is 2πn\frac{2\pi}{n}. The vertices are concyclic, and the polygon also has an incircle. The vertices of a regular nn-gon with the circumradius 11 can be (cos⁡2πkn,sin⁡2πkn)\left(\cos \frac{2\pi k}{n}, \sin \frac{2\pi k}{n}\right), for k=0,…,n−1k = 0, \dots, n - 1. With the side ss, the area is 14ns2cot⁡πn\frac{1}{4} n s^2 \cot \frac{\pi}{n}. Star polygons, such as the pentagram, are also called regular polygons. They are not convex.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

Show the path as a graph
Angle at a point, concept, no evidence levelAngle at a pointconceptConcyclic points, concept, no evidence levelConcyclic pointsconceptEuclidean distance, concept, no evidence levelEuclidean distanceconceptEuclidean plane, object, no evidence levelEuclidean planeobjectRegular polygon, definition, no evidence levelRegular polygondefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Angle at a point concept
  2. Concyclic points concept
  3. Euclidean distance concept
  4. Euclidean plane object
  5. Regular polygon definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Regular_polygon, lead section
  2. Web https://en.wikipedia.org/wiki/Regular_polygon, General properties
  3. Web https://en.wikipedia.org/wiki/Regular_polygon, Angles
About this page
Tag0V63D9V0FY
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Kinddefinition
WritingW1 stated
MSC51M20
DefinesRegular polygon
UsesEuclidean plane, Euclidean distance, Angle at a point, Concyclic points

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