Activity 4
Polygons

Polygons include any shapes constructed from segments that are simple, closed, and lie in the plane. Triangles are three-sided polygons. Triangles can be classified according to their sides and angles.

Triangle Classification

Equilateral		All sides have the same length.
Isosceles		Two sides have the same length.
Scalene		All sides have different lengths.
Acute		All angles are less than 90 degrees.
Obtuse		One angle is greater than 90 degrees.
Right		One angle is 90 degrees.

Use a 3 by 3 region of a geoboard to make all the possible triangles, recording each different triangle on the dot grids below. Describe each triangle by its sides and angles and write this below each dot grid.

Which classification categories were not included above?

________________ ________________ ________________

Two of these can be constructed on a 4 by 4 geoboard. Find each one, record on the dot grid, and label each one on the line below the grid.

Quadrilaterals may also be classified according to their angles and sides, as well as whether or not opposite sides are parallel.

Quadrilateral Classification

Parallelogram		Opposite sides are parallel.
Rectangle		Parallelogram with right angles.
Square		Rectangle with all sides same length.
Rhombus		Parallelogram with all sides same length.
Trapezoid		Exactly one pair of parallel sides.
Kite		Two pairs of adjacent sides of same length.
Concave		One interior angle is a reflex angle (>180°).

Use a 3 by 3 nail region of a geoboard to make all the possible quadrilaterals, recording each different quadrilateral on the dot grids below. Describe each quadrilateral as specifically as possible using the classification given above.

Which shape could not be made on the

3 by 3 nail geoboard? _______________

Sketch it on the 4 by 4 dot grid at the right.

Regular Polygons

Regular polygons are polygons that have all sides the same length and all angles the same measure. Each polygon is named according to the number of sides. The sum of the measures of the angles of a polygon can be found by separating the polygon into triangles. The measure of each angle of a regular polygon can be found by dividing the sum of the measures of the angles by the number of angles. Write the common name, and find both the sum of the measures of the angles and the measure of each angle of the regular polygons in the table below.

Shape
Name
Angle Sum
Each Angle
Shape
Name
Angle Sum
Each Angle

Use the information in the tables above to find the sum of the angles and the measure of each angle of a regular 100-sided polygon. Hint: imagine separating the 100-gon into triangles. How many triangles would there be?

Sum of Angles of Regular 100-gon = __________ Each Angle = __________

Generalize this information to write an expression for the sum of the angles and the measure of each angle of a regular n-sided polygon. Hint: imagine separating the n-gon into triangles. How many triangles would there be? Sum of Angles of Regular n-gon = __________ Each Angle = __________

Diagonals of Polygons

A diagonal of a polygon is a line segment that connects two vertices of the polygon that are not next to each other. Draw as many diagonals as possible for the hexagon below. How many are there? ___________

Determine the number of diagonals for each of the polygons below, either by drawing or by reasoning. Use the results to fill in the table below.

Shape
Sides
Diagonals
Shape
Sides
Diagonals

Use the results from the table to determine how many diagonals can be drawn on a regular polygon with 100 sides. Generalize to the case where the regular polygon has n sides.

Diagonals for 100-sided Regular Polygon = __________

Diagonals for n-sided Regular Polygon = ____________

Polygonal Regions

Draw a diagonal on the regular dodecagon below. This diagonal separates the dodecagon into 2 polygonal regions. Draw another diagonal that intersects the first diagonal at a point other than its endpoints. The dodecagon is now separated into 4 polygonal regions. What is the maximum number of polygonal regions that can be formed by adding a third diagonal to the dodecagon? Draw this diagonal. Draw a fourth diagonal so that the maximum number of polygonal regions is formed. Record your results in the table at the right of the dodecagon.

Diagonals Polygonal Regions

1 __________

2 __________

3 __________

4 __________

Repeat this process with the regular 1000000-gon shown below at the left. Record your results in the table below at the right and determine the values for 5, 6, 7, 8 diagonals. Generalize your results to n diagonals.

Diagonals Polygonal Regions

1 __________

2 __________

3 __________

4 __________

5 __________

6 __________

7 __________

8 __________

n ______________