Special Lines and Constructions of Regular Polygons A regular polygon with a center A is made up of congruent isosceles triangles with a principal angle A. The red line in the regular pentagon below is an axis of symmetry. Note that point A is at the center of the regular pentagon. To locate the center of a regular polygon, we will use the axis of symmetry that can be drawn from each vertex of the regular polygon. We will draw the five axes of symmetry to locate the center (A) of the regular pentagon. We can think of the axis of symmetry in two ways: 1. The right bisector of each side of a regular polygon is an axis of symmetry of the regular polygon. 2. The angle bisector of each side of each interior angle of a regular polygon is an axis of symmetry of the regular polygon. 1
A regular polygon s diagonal is a line segment joining two non-consecutive vertices of a regular polygon. The number of diagonals from a vertex depends on the number of sides (n). # of diagonals = n 3 where n is the number of sides of the regular polygon These are the 3 diagonals from point A in a regular hexagon. So, for a regular hexagon with n = 6, # of diagonals = n 3 = 6 3 = 3 This is what we saw in the above diagram. We mentioned at the beginning of this section that all regular polygons are made from congruent isosceles triangles. The number of congruent isosceles triangles are the same as number of sides that make the regular polygon. These are the 2 diagonals from point A in a regular pentagon. So, a regular pentagon is made up of 5 congruent isosceles triangles since a regular pentagon has 5 sides. In order to determine the central angle ( A), we use the following formula: A = n where n is the number of sides This is the 1 diagonal from point A in a square. Do you notice a trend? So, for a regular pentagon that has n = 5, A = n = 5 = 72 2
The 5 apothems for the regular pentagon from above are drawn in red below. Since the triangles are isosceles, we can determine all three angles in each triangle. You will use the apothem (a) to calculate the area (A) of a regular polygon. Finally, we will look at how to properly draw a regular polygon. Remember that the interior angles of a triangle add to 180 x+ x+ 72 = 180 + 72 = 180 = 180 72 To do this, we will use a ruler and a protractor, and our knowledge of the internal angles of a regular polygon. Let s say that we want to draw an equilateral triangle with sides that measure 4cm each. First, we use a ruler to draw a line 4cm long. = 108 x = 108 2 x = 54 Next, we remember that the interior angles of an equilateral triangle measures 60 each. All 5 triangles in a regular pentagon will always have congruent isosceles triangles with these angle measures. The altitude of each isosceles triangle from the central angle ( A) is called the apothem. 3
Now, we use a protractor and a ruler to draw a side that connects to vertex B at an angle of 60 and a length of 4cm. Example 1. In the regular polygon given below, identify the red line segment. Finally, we connect vertices A and C to complete the drawing of the equilateral triangle. Step 1: Refer to the notes on lines in a regular polygon. The red line in this regular polygon is a diagonal. We are ready to do more challenging examples. 4
2. How many diagonals can be drawn from vertex A in the regular hexagon below? 3. How many congruent isosceles triangles fit into a regular octagon? Step 1: Determine the number of sides that a regular octagon has. A regular octagon has 8 sides. This means that 8 congruent isosceles triangles fit into a regular octagon. Step 1: Use the formula for the number of diagonals with n = 6. # of diagonals = n 3 = 6 3 = 3 4. Determine the angle measures of the 10 congruent triangles that fit into a regular decagon. Step 1: Since n = 10, we can calculate one of the triangle s angles by dividing by 10. The 3 diagonal lines drawn from vertex A. A = n = = 36 10 5
Step 2: Determine the remaining two angles in the triangles. 5. Construct a square with sides measuring 2.5cm each by using a ruler and a protractor. Step 1: Using your ruler, draw a side measuring 2.5cm. The two remaining angles (x) are equal since this is an isosceles triangle. x+ x+ 36 = 180 Step 2: The interior angles of a square measure 90 each. Attach another side measuring 2.5cm to vertex B at a 90 angle to side AB. + 36 = 180 = 180 36 = 144 x = 144 2 x = 72 Step 3: Follow the same procedure from vertices A and C to complete the square. Step 3: Display the angle measures of one of the congruent triangles in a regular decagon. 6