4. A SQUARE is a rectangle having its adjoining
ABCD is a square.
AC and BD are diagonals.
O is the geometric center of the square.
AO = OC = OB = OD.
O is equidistant from all sides.
A square may be constructed if one side is
A polygon is a many-sided plane shape. It is said to
be regular if aIl sides are equal and irregular when they
are not. Only regular polygons are described here.
Triangles and quadrilaterals fit the description of a
polygon and have been covered previously. Three other
types of regular polygons are shown in the illustration.
Each one is inscribed in a circle. This means that all
vertices of the polygon lie on the circumference of the
Note that the sides of each of the inscribed polygons
are actually equal chords of the circumscribed circle.
Since equal chords subtend equal arcs, by dividing the
circumference into an equal number of arcs, a regular
polygon may be inscribed in a circle. Also note that the
central angles are equal because they intercept equal
arcs. This gives a basic rule for the construction of
regular polygons inscribed in a circle as follows:
To inscribe a regular polygon in a circle, create
equal chords of the circle by dividing the circumference
into equal arcs or by dividing the circle into equal central
Dividing a circle into a given number of parts has
been discussed, so construction should be no problem.
Since there are 360 degrees around the center of the
circle, you should have no problem in determining the
number of degrees to make each equal central angle.
What is the central angle used to inscribe a pentagon in
= 72° in each circle
Methods for Constructing Polygons
The three methods for constructing polygons de-
scribed here are the pentagon, the hexagon, and the
The PENTAGON is a method to develop the length
of a side and departs from the rule given. Radius PB has
been bisected to locate point O. Radius OC has been
used to swing an arc CE from the center O. E is the
intersection of arc CE with diameter AB. Chord CE is
the length of the side and is transferred to the circle by
arc EF using chord CE as radius and C as center.
The HEXAGON has been developed by dividing
the circumference into 6 equal parts.
The OCTAGON method has been developed by
creating central angles of 90° to divide a circle into 4
parts and bisecting each arc to divide the circumference
into 8 equal parts,