Difference between revisions of "Convex polygon"

(See also)
m
 
(6 intermediate revisions by 5 users not shown)
Line 1: Line 1:
{{WotWAnnounce|week=Sep 20-26}}
 
 
 
[[Image:convex_polygon.png|right]]
 
[[Image:convex_polygon.png|right]]
  
A '''convex polygon''' is a [[polygon]] whose [[interior]] forms a [[convex set]].  That is, if any 2 points on the [[perimeter]] of the polygon are connected by a [[line segment]], no point on that segment will be outside the polygon.
+
A '''convex polygon''' is a [[polygon]] whose [[interior]] forms a [[convex set]].  That is, if any 2 points on the [[perimeter]] of the polygon are connected by a [[line segment]], no point on that segment will be outside the polygon.  For example, every [[regular polygon]] is convex.
  
All [[internal angle]]s of a convex polygon are less than <math>180^{\circ}</math>. These internal angles sum to <math>180(n-2)</math> degrees.
+
All [[interior angle|interior angles]] of a convex polygon are less than <math>180^{\circ}</math>. Equivalently, all [[exterior angle|exterior angles]] are less than <math>180^{\circ}</math>.  The sum of the exterior angles of any convex polygon is <math>360^\circ</math> and the sum of the internal angles of a convex <math>n</math>-gon is <math>(n - 2)180^\circ</math>.
  
The [[convex hull]] of a set of points also turns out to be the convex polygon with some or all of the points as its [[vertices]].
+
The [[convex hull]] of a [[finite]] set of points is a convex polygon with some or all of the points as its [[vertex | vertices]].
 
 
The area of a regular [[n-gon]] of side [[length]] s is <math>\frac{ns^2*\tan{(90-\frac{180}{n})}}{4}</math>
 
  
 
== See also ==
 
== See also ==
 
* [[Concave polygon]]
 
* [[Concave polygon]]
 
* [[Convex polyhedron]]
 
* [[Convex polyhedron]]
* [[Concave polyhedron]]
 
  
 
{{stub}}
 
{{stub}}
  
 
[[Category:Definition]]
 
[[Category:Definition]]
 +
 +
[[Category:Geometry]]

Latest revision as of 21:10, 27 February 2020

Convex polygon.png

A convex polygon is a polygon whose interior forms a convex set. That is, if any 2 points on the perimeter of the polygon are connected by a line segment, no point on that segment will be outside the polygon. For example, every regular polygon is convex.

All interior angles of a convex polygon are less than $180^{\circ}$. Equivalently, all exterior angles are less than $180^{\circ}$. The sum of the exterior angles of any convex polygon is $360^\circ$ and the sum of the internal angles of a convex $n$-gon is $(n - 2)180^\circ$.

The convex hull of a finite set of points is a convex polygon with some or all of the points as its vertices.

See also

This article is a stub. Help us out by expanding it.

Invalid username
Login to AoPS