InterMath Dictionary

Polygon

 Adjacent Sides of a Polygon  Convex  
 Area of a Polygon  Curve  Irregular Polygon
   Diagonal  Perimeter
 Base of a Polygon  Exterior Angle  Polygon
 Circumscribed  Interior Angle  Regular Polygon
 Concave  Inscribed Polygon  Side of a Polygon


Circumscribed - A descriptor for a geometric figure that is drawn around and enclosing another geometric figure. A polygon is circumscribed about a circle if and only if each side of the polygon is tangent to (touches) the circle.
Diagonal - A line segment formed by connecting two nonadjacent vertices (i.e., not on the same side).
Area of a Polygon - A measure of how much surface is covered by a polygon. Areas are measured in square units.
Perimeter - The sum of the lengths of the sides of a figure.

 

Check for Understanding

Answer the following questions about polygon properties.

Enter each answer in the left column, then press tab

  Graded Response
a) Which of the following segments is a diagonal in hexagon ABGFED?      
b) What is the perimeter of polygon ABCDE ?     cm  
c) Which of the following circles circumscribes a polygon?    

Circumscribed Circles and Polygons in Your World

These quilts show circumscribed circles and polygons.

 

External Links to Extend Basic Understanding

More Definitions: Visit another online dictionary to clarify your understanding.
Perimeter Problems: Here you can find some practice problems with perimeter.
More Perimeter Problems: Here you can find more practice problems with perimeter.

Download: Download a game to practice with perimeter (note: Mac only).

 

Related Investigations on InterMath to Challenge Teachers

Diagonals in a Polygon: Consider the number of diagonals in a triangle, quadrilateral, pentagon, hexagon, heptagon, and octagon.
Round Robin Tournament
: Use the sides and diagonals in a regular polygon to create a round robin tournament
Apothem and Area: Explore the relationship between the apothem, perimeter, and area of a regular polygon.
Surrounding Squares
: Find a method that will construct a square that has the same area as the area of a given circle.
Inscribed in a Triangle
: Construct a circle inscribed in a triangle so that it will always remain inscribed in the triangle.


  A curve - Take a pencil and piece of paper and draw a path without lifting the pencil from the paper or retracing any part of the path, except to cross it. A polygonal curve - A curve that is entirely made up of line segments. (no arcs)
  A closed curve - A curve that begins and ends in the same location.   A polygonal closed curve - A polygon curve that begins and ends in the same location.
  A simple closed curve - A closed curve that does not cross itself.   A simple polygonal closed curve - A polygon closed curve that does not cross itself.

Polygon - A closed figure formed by three or more line segments.
Regular Polygon - When a polygon is equiangular (all angles are equal) and equilateral (all sides are equal) we say that it is regular. Regular polygons that we are familar with would be the equilateral triangle or the square.
Irregular Polygons - When a polygon's interior angles are not equal or its sides are not equal in length.

 

Did you know?

All polygons are simple, closed polygonal curves. The word "polygon" derives from the Greek words poly (many) and gonu (knee). So a polygon is a thing with many knees!

 

Regular Polygons in Your World

The eight-sided polygon picture in the tile below is regular. So are the six-sided polygons used to make the quilt.

 

Check for Understanding

How would you describe me? Give the most accurate response.

How would you describe me? Give the most accurate response.

 

External Links to Extend Basic Understanding

Further Investigations of Polygons: Investigate other features of regular polygons.
Polygons With LOTS of Sides
: How many sides does a polygon have to have before it turns into a circle? (or does it?)

 

Related Investigations on InterMath to Challenge Teachers

Vertex Angles in a Regular Polygon: Investigate how the vertex angle of a regular polygon relates to its number of sides.
Sum of Angles in a Polygon: Explore the relationship between the number of sides of a polygon and the sum of its interior angles.
Sum of Exterior Angles: Determine the sum of the exterior angles in any polygon.
Medial Polygons: Investigate the characteristics of a polygon that is generated by connecting the midpoints of consecutive sides of a polygon.
Apothem and Area: Explore the relationship between the apothem, perimeter, and area of a regular polygon.


See also Definitions Page: Polygon2 (information about specific polygons)

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Side of a Polygon- A straight line segment in a polygon.
Adjacent Sides of a Polygon - Two sides of a polygon that share a common vertex.

The line segments that form a polygon are called the sides of the polygon. A point where two sides meet is a vertex (plural form is vertices). Any two sides determine an interior angle of the polygon; an exterior angle is formed by a side and an adjacent side extended.

 

Given Polygon ABCD then its:

Sides are AB, BC, CD, CE, and EA.

Vertices are A, B, C, D, and E.

Interior angles: ex. angle BCD (green angle)

Exterior angles: ex. angle EDJ (yellow angle)

 

 

There are three parts to a polygon: the interior (green region), the exterior (yellow region), and the polygon itself (black region). The polygon and its interior make up a polygonal region.

 

Examples of Adjacent Sides

 

 

In triangle on the left, sides d and f are considered adjacent sides of the triangle ABC since they share the common vertex B. Can you ideniify other sides that are adjacent?

 

 

 

 

In the square on the right, sides f and g are considered adjacent sides of the square ABCD since they share the common vertex C.

 

 

 

Non-Examples of Adjacent Sides
In the square above, sides f and h are not adjacent sides since they do not share a common vertex. Likewise, sides e and g are not adjacent sides of the square.

 

Check for Understanding

Use quadrilateral ABCD to answer the following questions:

Enter the answer to part (a) in the left column, then press tab. Answer part (b) by making a selection from the menu.

 

Graded

Response

a) Identify a side that is adjacent to CD.    
b) What is the blue region called?  

 

External Links to Extend Basic Understanding

Further Investigations of Polygons: Investigate other features of regular polygons.
Flatland by Edwin A. Abbott: Read a classic book that imagines a two dimensional world, online!

 

Related Investigations on InterMath to Challenge Teachers

N-Sided Circle?: Compare the perimeters of polygons to the circumference of the circle.
Constructing quadrilaterals 1
: Given two parallel lines, form various quadrilaterals.
Integral sides
: Explore area and perimeter of quadrilaterals whose sides have integral length.
Double Trouble
: Modify components of a square and determine if a proportional relationship exists.
To Be or Not to Be
: Determine the necessary conditions to create a triangle.

See also Definitions Page: Triangles (general properties of triangles) and Triangles2 (information about specific triangles)

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Location, Location, Location
Interior Angle - An interior angle of a polygon is formed by two adjacent sides. Also, when a third line (a transversal) crosses two other lines, the angles formed in between the other two lines are called interior angles. (see alternate interior angles)
Exterior Angle - An exterior angle of a polygon is formed when you extend a side of a polygon. Also, when a third line (a transversal) crosses two other lines, the angles formed outside the other two lines are called exterior angles. (see alternate exterior angles)

 

An angle is an exterior angle of a polygon if and only if it forms a linear pair with one of the angles of the polygon.

 

 

An interior angle of a polygon is formed by two consecutive sides of a polygon. You will sometimes see an interior angle called a vertex angle.

 

 

Check for Understanding

Answer the following questions about interior and exterior angles related to triangle BCE.

 

Enter each answer in the left column, then press tab

 

Graded Response


a) Identify an exterior angle of the triangle. <
b) <ABC and what angle form a linear pair? <
b) What interior angle has a vertex that is point E? <

 

External Links to Extend Basic Understanding

Investigate Interior Angles of Polygons: What do all the interior angles of a polygon add up to?
Investigate Exterior Angles of Polygons: What do all the exterior angles of a polygon add up to?

 

Related Investigations on InterMath to Challenge Teachers

Remotely Interior: Compare exterior angles with remote interior angles.
Sum of Angles in a Polygon: Explore the relationship between the number of sides of a polygon and the sum of its interior angles.
Sum of Exterior Angles: Determine the sum of the exterior angles in any polygon.
Interior angles
: Explore the sum of interior angles of a quadrilateral.

See also Definitions Page: Angle


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Convex or Concave

Concave - A polygon that is not convex because one or more segments joining and two interior points does not lie completely in the polygon.
Convex - A polygon is convex if a segment joining any two interior points lies completely within the polygon.

A convex polygon can be determined using the following property: A line segment joining any two points inside the figure lies completely inside the figure. A non convex polygon is called a concave polygon. One way to remember this is to think of concave polygons being like caves.

Convex or Concave Polygons in Your World

Here are some examples of convex and concave polygons.

 

Check for Understanding

Enter the numerical values of the shapes that are convex, starting from the left. There may be more boxes than answers. Click on the check button after you have entered your answers.

Graded Response
       

 

External Links to Extend Basic Understanding

Ask Dr. Math: Dr. Math discusses convex and concave polygons.
Explore Convex Polygons: Investigate the relationships among polygons and their exterior and interior angles.

 

Related Investigations on InterMath to Challenge Teachers

Sum of Angles in a Polygon: Explore the relationship between the number of sides of a polygon and the sum of its interior angles.
Sum of Exterior Angles: Determine the sum of the exterior angles in any polygon.
Medial Polygons: Investigate the characteristics of a polygon that is generated by connecting the midpoints of consecutive sides of a polygon.
Quadrilaterals inscribed inside quadrilaterals
: Explore properties of quadrilaterals constructed using the midpoint of given quadrilaterals.
Quadrilaterals inscribed inside polygons
: How many quadrilaterals can be constructed using the vertices of polygons.

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Bases of Polygons

Base (of a polygon) - For two-dimensional figures, any side can be a base. But typically, the lower side is called the base. In three-dimensional figures, the base is typically a flat face that the figure can rest on.


Any side of a triangle can serve as a base. So any triangle has three bases.

 

A trapezoid is a quadrilateral with at least one pair of parallel sides (indicated below in red). These parallel sides in a trapezoid are called bases.

 

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Inscribed Polygon - A polygon is inscribed in a circle if and only if each of its vertices lie on the circle.

Triangle ABC, quadrilateral DEFG, and pentagon HIJKL are each inscribed in a circle. Notice for each inscribed polygon, all the vertices lie on the circle and the polygon lies entirely inside the circle. The circles that surround the polygons are called circumcircles.

 

Check for Understanding

Which of the following polygons is inscribed in a circle?

Enter your answer in the left column, then press tab

Graded Response
 
 

 

External Links to Extend Basic Understanding

Angles and Triangles in a Circle: Investigate properties of inscribed triangles.

 

Related Investigations on InterMath to Challenge Teachers

N-Sided Circle?: Compare the perimeters of polygons to the circumference of a circle.
Inscribed Parallelogram
: What can you tell about a parallelogram that is inscribed in a circle?
Half is much may be right
: Explore triangles inscribed in semicircles.
Triangles inscribed inside triangles
: Explore properties of triangles constructed using the midpoint of given triangles.
Pentagram Triangles
: How many triangles are formed?


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