CHAPTER 5
What is a convex polygon simple definition?
A convex polygon is a closed figure where all its interior angles are less than 180°, and the
vertices point outwards. The term convex is used to refer to a shape that has a curve or a
protruding surface. In other words, all the lines across the outline are straight and point
outwards.
A convex polygon is a closed shape with no vertices pointing inward. When a line crosses the convex
polygon, it intersects the two sides of the polygon at most.
Any polygon, such as a pentagon, hexagon, etc., whose vertices point outside the center of the
polygon is convex.
Are all polygon convex?
Every polygon is either convex or concave. The difference between convex and concave polygons
lies in the measures of their angles. For a polygon to be convex, all of its interior angles must be less
than 180 degrees. Otherwise, the polygon is concave.
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The sum of the measures of the four angles of any quadrilateral is 360°.
The sum of the measures of the five angles of any pentagon is 540°.
The sum of the measures of the n interior angles of an n-gon is (n - 2)180-degrees.
For any polygon, the sum of the measures of a set of exterior angles is 360°.
You can find the measure of each interior angle of an equiangular n-gon by using either of
these formulas: (n - 2) x 180-degrees / n
EXTERIOR ANGLE CONJECTURE FOR QUADRILATERALS - PROOF
EXTERIOR ANGLE CONJECTURE FOR POLYGON - PROOF
STATEMENT
a+p=180, b+t=180, c+s=180, d+r=180, e+q=180
REASON
Linear Pair Conjecture
p+t+s+r+q = 540
Poly Sum Conjecture (n-2)180
a+p+b+t+c+s+d+r+e+q = 900
Addition Property Equality (180x5)
a+b+c+d+e = 360
Subtraction Property of Equality (900-540)
The measure of 1 exterior angle of an equiangular polygon with n sides is:
A polygon's sum of exterior angles is always equal to 360 degrees. Therefore, for all equiangular
polygons, the measure of one exterior angle is equal to 360 divided by the number of sides in
the polygon.
360/n