REGULAR AND CONVEX POLYGONS
A polygon is a closed plane figure formed by three or
more sides. These sides only intersect at endpoints.
POLYGONS
External elements in the polygon:
1)Sides: line segments that delimit and shape the polygon. They are
usually named according to the vertices that delimit them.
2)Vertices: the points where two straight sides coincide. They are named
with a capital letter.
°.
POLYGONS
External elements in the polygon:
3)Angles: the opening between two consecutive sides. They can be:
Internal: the angle formed by two adjacent sides.
External: the angle formed between the extension of one of its sides and its
adjacent side.
They are supplementary angles with their corresponding internal angles.
The sum of all external angles is 360
°.
POLYGONS
Internal elements in the polygon:
Diagonals: segments joining two angles or non-adjacent
vertices.
Regular polygons have:
Center: the center point measuring the same distance to all vertices or
all sides: O.
Apothem: line dimension joining the center to each midpoint of the
sides.
Radius: segment that goes from the center to each vertex of the polygon.
Central angle: it is formed by two segments that start from the center
towards two consecutive vertices.°.
Each polygon has a special name, which depends on the number of
sides.
REGULAR AND CONVEX POLYGONS
A polygon is regular when all its angles and sides are
equal (otherwise it is irregular).
REGULAR AND CONVEX POLYGONS
A convex polygon is defined as a polygon with all its
interior angles smaller than 180°. This means that all
the vertices of the polygon will point outwards, away
from the interior of the shape.
A regular polygon is always convex, but not viceversa.
Regular and convex polygon
.
REGULAR AND CONVEX POLYGONS
The sum of the measures of the exterior angles
of any convex polygon is 360°.
The sum S of the measures of the interior
angles of a convex polygon having n sides is:
𝐒 = 𝟏𝟖𝟎(𝒏 − 𝟐)
REGULAR AND CONVEX POLYGONS
EXAMPLE:
Find the measure of each interior angle and each exterior
angle of a regular dodecagon (a twelve-sided polygon).
Since this is a regular dodecagon, the 12 exterior angle
measures are identical and add up 360°.
360°
𝑀𝑒𝑎𝑠𝑢𝑟𝑒 𝑜𝑓 𝑒𝑎𝑐ℎ 𝑒𝑥𝑡𝑒𝑟𝑖𝑜𝑟 𝑎𝑛𝑔𝑙𝑒 =
= 30°
12
REGULAR AND CONVEX POLYGONS
Since a regular polygon is always convex, the interior
angles are defined by the formula:
S = 180(𝑛 − 2)
S = 180(12 − 2)
S = 1800
Because the 12-gon is regular, each of its
12 interior angles has the same measure:
1800
𝑀𝑒𝑎𝑠𝑢𝑟𝑒 𝑜𝑓 𝑒𝑎𝑐ℎ 𝑖𝑛𝑡𝑒𝑟𝑖𝑜𝑟 𝑎𝑛𝑔𝑙𝑒 =
= 150°
12
INTERNAL, EXTERNAL ANGLES AND SUM OF ANGLES:
INTERNAL, EXTERNAL ANGLES AND SUM OF ANGLES:
PERIMETERS AND AREAS OF POLYGONS
The perimeter, both for regular and irregular polygons, is
constituted by the contour of the figure, so to know it all of
its sides must be found. The unit of measurement is
length.
The area of a figure is the amount of space that is
delimited by the sides, generating a surface. It is
measured in quadratic units of length, and like the
perimeter, formulas are required, which vary depending on
the figure.
FORMULAS
FORMULAS
FORMULAS