Remember that "regular" polygons have all sides congruent and all angles congruent. Regular polygons have a center and a radius (coinciding with their circumscribed circle), and the distance from the center perpendicular to any side is called its apothem.
Regular Pentagon |
The apothem of a regular polygon is a line segment from the center of the polygon perpendicular to any side of the polygon. Triangle DOC is an isosceles triangle, making the apothem the altitude of this triangle and the median of this triangle (going to the midpoint P.) The apothem is also the radius of the inscribed circle.
The apothem can be used to determine area:
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Area of a
REGULAR polygon |
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(where a = apothem and p = perimeter) |
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Here are the more popular regular polygons.
(Remember: The formula for each interior angle of a regular polygon is [180(n - 2)] / n where n = the number of sides.)
Equilateral Triangle
interior angle = 60º
Special 30º-60º-90º Δs
at work!
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Square
interior angle = 90º
Special 45º-45º-90º Δs
at work!
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Regular Pentagon
interior angle = 108º
Must use trig. to
work in this triangle.
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Regular Hexagon
interior angle = 120º
Special 30º-60º-90º Δs
at work!
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Regular Octagon
interior angle = 135º
Must use trig. to
work in this triangle.
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Using the strategy of partition or dissection, the areas of these regular polygons can be found by adding together the areas of all of the congruent triangles formed by the central angles and each side of the polygon.
The number of such triangles = the number of sides of the polygon.
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In addition to using the strategy of partitioning or dissecting, 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. |
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