Sample Problems:

Problem: ABCD is a parallelogram and FBED is a square. What is the perimeter of ABCD if BD (not shown) equals 8√2 and AB = 10?

Solution: Since FBED is a square, BF = FD and Triangle BFD is a 45-45-90 triangle.

      

      Since Triangle BFD is a 45-45-90 triangle, s = 8. Then AF can be found with the Pythagorean Theorem.

           a² + b² = c² --> a² + 82 = 102 --> a² = 36 --> a = 6 --> AF = 6

      

Perimeter = AB + BC + CD + DA = 10 + 14 +10 + 14 = 48

Problem: A 6 sided regular polygon (hexagon) is inscribed in a circle of radius 10 cm, find the length of one side of the hexagon. 

Solution:

• Angle AOB is given by
  
  angle (AOB) = 360^o / 6 = 60^o
• Since OA = OB = 10 cm, triangle OAB is isosceles which gives
  
  angle (OAB) = angle (OBA)
• So all three angles of the triangle are equal and therefore it is an equilateral triangle. Hence
  
  AB = OA = OB = 10 cm. 

Polygons and Quadrilateral

Polygons are a very useful graphical tool. Solid three dimensional shapes can easily be approximated with a few polygons, and, when good shading and texturing are applied, they can look reasonably realistic. They can be drawn quickly and take up very little storage space.


Convex Polygon: Your normal standard run of the mill shape. Triangles, squares, hexagons all fit into this category. The internal angles must all be less than 180 degrees.
Concave Polygon: Concave polygons go back on themselves. They can contain inlets and fjords. They all have at least 4 sides.
Complex Polygon: These are all the rest. They can do anything. They can have holes, they can cross themselves, as many times as they like.

Types of Polygon


There are many types of polygons.


In any polygon a pair of sides that are inclined to each other meet at a point. This point is called a vertex of the polygon. In the polygon there are as many vertices as there are sides.


The triangle has 3 sides and 3 vertices.

 

The quadrilateral has 4 sides and 4 vertices.





The pentagon has 5 sides and 5 vertices.





The hexagon has 6 sides and 6 vertices.







Polygons are all around. Take a close look around you in your house, at the work place, or in an amusement park, you will find many real life examples of polygons.






Angle measures of Quadrilateral

• Four sides (or edges)
• Four vertices (or corners).
• The interior angles add up to 360 degrees:  

Measure the Angles, They should be 360°


Polygons in Nature

There are polygons found in nature everywhere, 

Rocks





Flower