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WHAT IS A POLYGON? - DEFINITION, SHAPES & ANGLES

Lesson Transcript
Cite this lesson
Instructor
Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has been teaching math for
over 9 years. Amy has worked with students at all levels from those with special
needs to those that are gifted.

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Expert Contributor
Elaine Chan

Dr. Chan has taught computer and college level physics, chemistry, and math for
over eight years. Dr. Chan has a Ph.D. in Chemistry from U. C. Berkeley, an M.S.
Physics plus 19 graduate Applied Math credits from UW, and an A.B. with honors
from U.C .Berkeley in Physics.

View bio
Polygons are defined as any 2-dimensional shapes constructed with straight
lines. Learn about the common and uncommon polygon shapes and how to design them
using interior and exterior angles. Updated: 09/22/2021
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DEFINITION OF A POLYGON

A polygon is any 2-dimensional shape formed with straight lines. Triangles,
quadrilaterals, pentagons, and hexagons are all examples of polygons. The name
tells you how many sides the shape has. For example, a triangle has three sides,
and a quadrilateral has four sides. So, any shape that can be drawn by
connecting three straight lines is called a triangle, and any shape that can be
drawn by connecting four straight lines is called a quadrilateral.



Polygons



All of these shapes are polygons. Notice how all the shapes are drawn with only
straight lines? This is what makes a polygon. If the shape had curves or didn't
fully connect, then it can't be called a polygon. The orange shape is still a
polygon even if it looks like it has an arrow. All the sides are straight, and
they all connect. The orange shape has 11 sides.

I've mentioned a few polygons and have shown you a few common shapes. Here is a
list of those in addition to several more:



Shape # of Sides Triangle 3 Square 4 Rectangle 4 Quadrilateral 4 Pentagon 5
Hexagon 6 Heptagon 7 Octagon 8 Nonagon 9 Decagon 10 n-gon n sides

The last entry includes the general term for a polygon with n number of sides.
Polygons aren't limited to the common ones we know but can get pretty complex
and have as many sides as are needed. They can have 4 sides, 44 sides, or even
444 sides. The names would be 4-gon, or quadrilateral, 44-gon, and 444-gon,
respectively. An 11-sided shape can be called an 11-gon.

6:09











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 * 0:01 Definition of a Polygon
 * 1:54 Regular Polygons
 * 3:02 Angles of Regular Polygons
 * 5:35 Lesson Summary

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REGULAR POLYGONS

A special class of polygon exists; it happens for polygons whose sides are all
the same length and whose angles are all the same. When this happens, the
polygons are called regular polygons. A stop sign is an example of a regular
polygon with eight sides. All the sides are the same and no matter how you lay
it down, it will look the same. You wouldn't be able to tell which way was up
because all the sides are the same and all the angles are the same.

When a triangle has all the sides and angles the same, we know it as an
equilateral triangle, or a regular triangle. A quadrilateral with all sides and
angles the same is known as a square, or regular quadrilateral. A pentagon with
all sides and angles the same is called a regular pentagon. An n-gon with sides
and angles the same is called a regular n-gon.



Regular polygons



Here is a regular triangle, a regular quadrilateral, and a regular pentagon. Do
you see how all the sides are the same and no matter how you flip it, it will
look the same?


ANGLES OF REGULAR POLYGONS

Regular polygons also have two different angles related to them. The first is
called the exterior angle, and it is the measurement between the shape and each
line segment when you stretch it out past the shape.



The exterior angle



However many sides a polygon has is the same number of exterior angles it has.
So, a pentagon with five sides has five exterior angles. A hexagon will have six
exterior angles and so on. For regular polygons, we can figure out the
measurement of the exterior angle, but for polygons that aren't regular, we
can't. Here is the formula for regular polygons:



The exterior angle formula



The n stands for the number of sides the polygon has. So, a pentagon has
exterior angles that measure 360 / 5 = 72 degrees.

The second angle is called the interior angle, which is the supplementary angle
to the exterior angle. This means that the interior angle together with the
exterior angle will add up to 180 degrees.



The interior angle



You can also say that the interior angle is the measurement of each corner of
the polygon. Here is the formula for the interior angle:



The interior angle formula



The second formula is the same as the first, just rearranged. Don't worry about
how we got there right now; just remember one or the other, and you will be
okay. The second one is the more commonly seen in the math world. Let's look at
an example. For our pentagon with five sides, using the first equation gives us
180 - 360 / 5 = 180 - 72 = 108 degrees. Using the second equation, we get (5 -
2) * 180 / 5 = 3 * 180 / 5 = 540 / 5 = 108 degrees. Both formulas will give us
the same answer. Choose the formula that is easier for you to remember.


LESSON SUMMARY

Polygons are all around us. Who of us has ever seen a triangle or a square? A
polygon is defined as a 2-dimensional shape with straight sides. Regular
polygons have sides and angles that are all the same. While you can find the
measurements of the exterior and interior angles of regular polygons, you can't
with polygons that aren't regular.


LEARNING OUTCOMES

After this lesson, you should be able to:

 * Define polygon and regular polygon
 * Identify examples of polygons and regular polygons
 * Explain how to find the exterior and interior angles of regular polygons

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Additional Activities


SUM OF POLYGON INTERIOR ANGLES




Each segment that forms a polygon is a side of the polygon. The endpoint of two
sides is a vertex of the polygon. A segment that connects any two nonconsecutive
vertices is a diagonal.

A polygon is concave if any part of a diagonal contains points in the exterior
of the polygon. If no diagonal contains points in the exterior, then the polygon
is convex. A regular polygon is always convex.

To find the sum of the interior angle measures of a convex polygon, draw all
possible diagonals from one vertex of the polygon. This creates a set of
triangles. The sum of the angle measures of all the triangles equals the sum of
the angle measures of the polygon.





PLEASE FILL IN THE BLANKS




For the triangle, the number of sides = _____. The number of triangles =_____.
The sum of interior angle measures = ( _____ ) 180 degrees.




For the quadrilateral, the number of sides = _____. The number of triangles
=_____. The sum of interior angle measures = ( _____ ) 180 degrees.




For the pentagon, the number of sides = _____. The number of triangles =_____.
The sum of interior angle measures = ( _____ ) 180 degrees.




For the hexagon, the number of sides = _____. The number of triangles =_____.
The sum of interior angle measures = ( _____ ) 180 degrees.




For the n-gon, the number of sides = _____. The number of triangles =_____. The
sum of interior angle measures = ( _____ ) 180 degrees.




In each convex polygon, the number of triangles formed is two less than the
number of sides n. So the sum of the angle measures of all these triangles

is (n - 2 ) 180 degrees.

The measure of each interior angle of a regular n-gon is (n-2)180/n degrees.


ANSWERS:

Triangle: 3,1,1

Quadrilateral: 4,2,2

Pentagon: 5,3,3

Hexagon: 6,4,4

n-gon: n, n-2, n-2


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Math / Geometry: High School
Geometry: High School

14 chapters | 145 lessons

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Ch 1. High School Geometry: Foundations of...
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Ch 2. High School Geometry: Logic in...
 * Go to High School Geometry: Logic in Mathematics

Ch 3. High School Geometry: Introduction to...
 * Go to High School Geometry: Introduction to Geometric Figures

Ch 4. High School Geometry: Properties of...
 * Go to High School Geometry: Properties of Triangles

Ch 5. High School Geometry: Triangles,...
 * Go to High School Geometry: Triangles, Theorems and Proofs

Ch 6. High School Geometry: Parallel Lines...
 * Go to High School Geometry: Parallel Lines and Polygons

Ch 7. High School Geometry: Similar Polygons
 * Ratios and Proportions: Definition and Examples 5:17
 * Geometric Mean: Definition and Formula 5:15
 * Angle Bisector Theorem: Definition and Example 4:58
 * Solving Problems Involving Proportions: Definition and Examples 5:22
 * Similar Polygons: Definition and Examples 8:00
 * The Transitive Property of Similar Triangles 4:50
 * Triangle Proportionality Theorem 4:53
 * Constructing Similar Polygons 4:59
 * Properties of Right Triangles: Theorems & Proofs 5:58
 * The Pythagorean Theorem: Practice and Application 7:33
 * The Pythagorean Theorem: Converse and Special Cases 5:02
 * Similar Triangles & the AA Criterion 5:07
 * What is a Polygon? - Definition, Shapes & Angles 6:08
 * Go to High School Geometry: Similar Polygons

Ch 8. High School Geometry:...
 * Go to High School Geometry: Quadrilaterals

Ch 9. High School Geometry: Circular Arcs and...
 * Go to High School Geometry: Circular Arcs and Circles

Ch 10. High School Geometry: Conic...
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Ch 11. High School Geometry: Geometric...
 * Go to High School Geometry: Geometric Solids

Ch 12. High School Geometry: Analytical...
 * Go to High School Geometry: Analytical Geometry

Ch 13. High School Geometry:...
 * Go to High School Geometry: Probability

Ch 14. High School Geometry: Introduction to...
 * Go to High School Geometry: Introduction to Trigonometry

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