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Regular Polygon
Introduction:
It is often necessary to determine the dimensions of a regular polygon inscribed in or circumscribed
about a given circle, or to determine the size of a circle that can be inscribed in or circumscribed about a
given polygon. Such problems are readily solved by trigonometry and some of them may be solved by
geometry. The practical applications of polygons are the gears.
Regular Polygon:
A plane figure bounded by a number of straight lines is called a polygon. A regular polygon is the
one in which all sides and all the internal angles are equal. A regular polygon has a point O, inside it, called
its center which is equidistant from all its vertices and sides. A convex polygon is the one in which no
angle is greater than two right angles, i.e. its has no reflex or re-entrant angle. (Regular polygons are convex
polygons). From the above definitions, triangles and quadrilaterals are also polygons but generally we
define the polygon as the figure bounded by more than four straight lines. The sum of the bounding sides of
the polygon is called theperimeter of the polygon.
Circumscribed and Inscribed Circle
Circumscribed Circle:
If a polygon is drawn in a circle so that every corner of the polygon lies on the circle, the polygon is
called an inscribed polygon and the circle is called the circumscribed circle. In the figure, is
the center of the polygon and
is the radius of the circumscribed circle and id denoted by .
Inscribed Circle:
If a polygon is drawn outside a circle so that every side of the polygon touches the circle, the polygon
is called the circumscribed polygon and the circle is called the inscribed circle. In the figure,
is the
radius of the inscribed and is denoted by .
If the polygon is regular then the center of the circle is called is also the center of the polygon.
Properties of a Regular Polygon
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The sides and interior angles of a regular polygon are all equal.
The bisectors of the interior angles of a regular polygon meet at its center.
The perpendiculars drawn from the center of a regular polygon to its sides are all equal.
The lines jointing the center of a regular polygon to its vertices are all equal.
The center of a regular polygon is the center of both the inscribed and circumscribed circles.
Straight lines drawn from the center to the vertices of a regular polygon, divides it into as many
equal isosceles triangles as there are sides in it.
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The angle of a regular polygon of
sides
.
Detail of Sides of Polygon:
There is no theoretical limit to the number of sides of a polygon, but only those with twelve or less
are commonly met with. The names of polygons which are mostly in use as follows:
Number of sides
Polygon Name
Pentagon
Hexagon
Heptagon
Octagon
Nonagon
Decagon
gon
Example:
The perimeter and area of a regular polygon are respectively, equal to those of a square of sides .
Find the length of the perpendicular from the center of a regular polygon to any of its sides.
Solution:
Perimeter of a regular polygon = perimeter of square
A regular polygon can be divided into congruent triangles having common vertex at the center of the
polygon. The number of these congruent triangles is the same as that of its sides.
Area of one such triangle
any side of polygon.
sides of polygon
length of perpendicular from the center to
Area of one such triangle
,
being length of perpendicular
Area of Polygon = Sum of areas of all such triangles
Perimeter of polygon
--- (1)
Now, Area of Polygon = Area of Square
(given) --- (2)
From (1) and (2), we have
Hence, the length of perpendicular is
.
Area of a Regular Polygon 1
Area of a Regular polygon of n sides
when the Length of one side is given:
Let
, be the length of a side of a regular polygon of
polygon.
As there are sides, similar triangles are formed.
Let
be one of the
triangles, then
and
Area of the regular polygon
But
Or
,
area of
sides and
be the center of the
Or
Area of the regular polygon
Also, perimeter of the polygon
Example:
Find the cost of carpeting an octant floor with sides measuring
square meter.
Solution:
Here
,
, if the carpet costs Rs.
Area
Square meter
Square Meter Approximate.
Cost of carpeting
Rupees
Area of a Regular Polygon 2
Area of a Regular polygon of n sides when the radius r of the inscribed circle is given:
Since
Area of the polygon
area of the
per
But
,
and
Hence, area of the regular polygon
Also, perimeter of the polygon
(Putting for
Example:
A regular octagon circumscribed a circle of
Solution:
Here
,
Area of the polygon
from above)
radius. Find the area of the octagon.
Square meter approx.
Area of a Regular Polygon 3
Area of a Regular polygon of n sides when the radius r of the circumscribed circle is given:
Let
be the radius of the circumscribed circle.
Area of the polygon
area of
But Area of
Also, Perimeter of the polygon
But
Also
Perimeter
Or
(As
Example:
A regular decagon is inscribed in a circle, the radius of which is
decagon.
Solution:
Here
,
Area of decagon
Square cm
)
. Find the area of the