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Remember that "regular" polygons have all sides congruent (equal) and all angles congruent. Regular polygons have a center and a radius. This center and radius are also the center and radius of a circle circumscribed about the outside of the polygon.
polycircle
def
An apothem of a regular polygon is a line segment from the center of the polygon perpendicular to any side of the polygon.

Regular Pentagonpolyparts
The apothem of a regular polygon can be drawn to any side the polygon. Since is is perpendicular to the side, it makes a right angle with the side. Triangle DOC is an isosceles triangle (since the radii are equal), making the apothem the altitude (height) of this triangle. The apothem is also the radius of a circle inscribed inside the polygon.
The apothem can be used to determine area:
apothemlong

areapolyN

Area of a
REGULAR polygon
apothemformula
(where a = apothem and p = perimeter)

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Using the strategy of partitioning (decomposing),
the areas of these regular polygons can be found
by adding together the areas of all of the congruent triangles
formed by the radii drawn to each vertex.

Since the side of the polygon is a side of the triangle,
we know that ...
the number of such triangles = the number of sides of the polygon.
dissectingpolys
This observation regarding the triangles and the polygon
leads to the formula for the polygon's area:

apothemperimeterformula

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hint
In addition to using the strategy of partitioning or decomposing, it may also be possible to graph the polygon on a set of coordinate axes and determine the area using coordinate geometry techniques. A "grid" method may be useful if sufficient information is known.


Given hexagon ABCDEF with A(1,6), B(3,9), C(6,9),
D(8,6), E( 6,3), and F(3,3).

Find the number of square units in the area of ABCDEF.

Solution: The "box method" will work nicely.
ABox = AΔ1 + AΔ2 + AΔ3 + AΔ4 + Apoly

42 = 3 + 3 + 3 + 3 + Apoly

Apoly= 30 square units

polygrid
The "box method" will work with both regular and irregular polygons if sufficient information is given..


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