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Polygon Properties: Regular, Irregular, Area, Perimeter, & More

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Properties of Polygons
This section describes and compares the properties of regular and
irregular polygons.
Introduction to
Polygons
A polygon is a closed twodimensional shape with straight
sides. Polygons can have any
number of sides, and each side
must intersect with exactly two
other sides.
Regular Polygons
A regular polygon is a polygon
with all sides and angles
congruent. Examples of regular
polygons include equilateral
triangles, squares, and regular
hexagons.
Regular Polygons
A regular polygon is a polygon in which all sides are of equal length and all angles
are of equal measure.
Examples of regular polygons include triangles, squares, pentagons, hexagons,
and so on. Regular polygons have many interesting properties, such as:
•
•
•
•
All
interior angles are equal
•
All exterior angles are equal
The sum of the interior angles is equal to (n-2) times 180 degrees, where n is the
number of sides The measure of each interior angle is given by (n-2) times 180 degrees
divided by n
The measure of each exterior angle is given by 360 degrees divided by n
Irregular Polygons
Irregular polygons are polygons that do not have congruent sides or
angles. They can have any number of sides and angles, and their sides
can be of different lengths.
Unlike regular polygons, irregular polygons cannot be classified by a
single formula for calculating their perimeter or area. Instead, the
perimeter and area must be calculated by adding up the lengths of the
sides or using more complex formulas.
Angles in Polygons
Interior Angles
The sum of the interior angles in a polygon
can be calculated using the formula (n-2) x
180, where n is the number of sides in the
polygon. For example, a triangle has 3 sides,
so the sum of its interior angles is (3-2) x
180 = 180 degrees. A hexagon has 6 sides,
so the sum of its interior angles is (6-2) x
180 = 720 degrees.
Exterior Angles
The exterior angle of a polygon is
the angle formed by a
side of the polygon and the
extension of an adjacent
side. The sum of the exterior
angles in a polygon is
always 360 degrees. For example,
a regular pentagon
has 5 sides , so each exterior
angle measures 72 degrees
(360 / 5).
Perimeter of
Polygons
Regular Polygons
The perimeter of a regular
polygon can be found by
multiplying the length of one
side by the number of sides.
Irregular Polygons
The perimeter of an
irregular polygon can be
found by adding the
lengths of all its sides.
Area of Polygons
Regular Polygons
The area of a regular polygon can be calculated using the formula A
= (1/2) x ap, where a is the apothem (distance from the center to the
midpoint of a side) and p is the perimeter.
Irregular Polygons
The area of an irregular polygon can be calculated by dividing it
into smaller regular polygons and then summing their areas.
Congruent Polygons
Congruent polygons are polygons that have the same shape and size.
To show that two polygons are congruent, we use the symbol
≅
. For example, if polygon ABCD is congruent to
polygon
ABCDEFGH,
EFGH we write:
≅
In order for two polygons to be congruent, all corresponding sides and
angles must be equal. This means that if we know the measures of the sides
and angles of one polygon, we can use that information to find the measures
of the sides and angles of a congruent polygon.
Similar Polygons
Similar polygons are polygons that have the same shape but not necessarily the same size. This
means that the corresponding angles of similar polygons are congruent, and the corresponding sides
are proportional.
Proportions of Sides
To determine whether two polygons are similar, we need to compare the ratios of their corresponding
sides. If the ratios are equal, then the polygons are similar.
Applications
Similar polygons are used in many real-world applications, such as map-making, architecture, and
engineering. For example, architects use similar polygons to design buildings that have the same
shape but different sizes. Engineers use similar polygons to design structures such as bridges and
towers that are strong and stable.
Applications of Polygons
• Architecture: Polygons are widely used in architectural designs to create unique
•
and aesthetically pleasing structures. Examples include the Sydney Opera House
and the Guggenheim Museum Bilbao.
• Engineering: The properties of polygons are important in engineering applications
•
such as designing bridges, tunnels, and dams. Engineers use polygons to
determine the strength and stability of these structures.
Art and Design: Polygons are used in graphic design and digital art to create
intricate patterns and shapes. They are also used in 3D modeling and animation to
create realistic objects and environments.
Geography: Polygons are used in geography to represent land masses and bodies
of water on maps. They are also used to calculate the area and perimeter of these
regions.
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