The Diagonal
A diagonal of a polygon is a line segment that connects two nonconsecutive vertices.

An irregular quadrilateral with two diagonals; An irregular hexagon with diagonals
  1. Click on the link below to explore the diagonals of a polygon by dragging the orange dot to change the size of the polygon and click on the more or less link under "Number of Sides" to change the polygon. Observe several different polygons counting the number of diagonals of each.

    This activity might not be viewable on your mobile device.For this interactive activity, assistance may be required. Diagonals of a Polygon


    Source: Diagonals of a Polygon, Math Open Reference


  2. After observing the diagonals of several different polygons fill in the blanks of the chart in your notes.
    3. Polygon Number of Sides Number of Diagonals
    Quadrilateral    
    Pentagon    
    Hexagon    
    Octagon    
    Decagon    
    n-gon    
    Interactive popup. Assistance may be required.

    Click to check the answers to the first table.

    Polygon Number of Sides Number of Diagonals
    Quadrilateral 4 2
    Pentagon 5 5
    Hexagon 6 9
    Octagon 8 20
    Decagon 10 35
    n-gon n n(n – 3) 2
    Close Pop Up
  3. Change the polygon to irregular by clicking on the Make Irregular link in the upper left hand corner. Change the number of sides and observe the diagonals. Note the differences in the diagonals when the polygon is convex vs. concave.
  4. After observing the diagonals of concave and convex polygons with several different sides fill in the blanks of the chart in your notes.
    5. Polygon Number of Sides Number of Diagonals

    All the diagonals are always inside the polygon
    Yes/ No
    Quadrilateral      
    Pentagon      
    Hexagon      
    Interactive popup. Assistance may be required.

    Click to check the answers to the second table.

    Polygon Number of Sides Number of Diagonals

    All the diagonals are always inside the polygon
    Yes/ No
    Quadrilateral 4 2 No
    Pentagon 6 9 No
    Hexagon 5 5 No
    Close Pop Up
  5. Interactive exercise. Assistance may be required. Click on the four correct statements/conjectures below. The true statements will change to bold text.

  6. Review the correct conjectures. Write a statement in your notes describing your conclusion.

    Interactive popup. Assistance may be required.

    Click here to see a true statement on conjectures.

    Convex:
    All the diagonals of a convex polygon are located inside of the polygon.

    Concave:
    If at least one of the diagonals is located outside of the polygon the polygon is concave.

    n-gon:
    An n-gon has n sides and   n (n-3) 2  diagonals. Close Pop Up