1. Section 7.3 Regular Polygons and Area
A regular polygon is both
equilateral and equiangular.
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2. To construct a circle inscribed in a square:
Construction 1
To construct a circle inscribed in a square:
The Center O must be the point of concurrency of the angle bisectors of the square.
Construct the angle bisectors of two adjacent vertices to find O.
From O, construct OM perpendicular to a side.
The length of OM is the radius.
Diagrams p. 357 Figure 7.32
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3. Given a regular hexagon, construct a circumscribed  X.
Construction 2
Given a regular hexagon, construct a circumscribed  X.
The center of the circle must be equidistance from each vertex of the hexagon.
Construct two perpendicular bisectors of two consecutive sides of the hexagon
The Center X is the point of concurrency of these bisectors.
The length from X any vertex is the radius of the circle.
Diagrams p. 357 Figure 7.33
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4. A radius of a regular polygon is any line segment that joins the
Definitions
The center of a regular polygon is the common center for the inscribed and circumscribed circles of the polygon.
A radius of a regular polygon is
any line segment that joins the
center of the regular polygon to
one of its vertices.
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5. An apothem of a regular polygon is any line segment drawn from the center of that polygon perpendicular to one of the sides.
A central angle of a regular polygon is an angle formed by two consecutive radii of the regular polygon.
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6. Theorems
Theorem 7.3.1: A circle can be circumscribed about (or inscribed in) any regular polygon.
Theorem 7.3.2: The measure of the central angle of a regular polygon of n sides is given by c = 360/n.
Theorem 7.3.3: Any radius of a regular polygon bisects the angle at the vertex to which it is drawn.
Theorem 7.3.4: Any apothem of a regular polygon bisects the side of the polygon to which it is drawn.
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7. Area of a Regular Polygon
Theorem 7.3.5:
The area A of a regular polygon whose apothem has length a and whose perimeter P is given by: A = ½ aP.
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