Geometry : Unit #2 - Geometric Constructions : Lesson #10

Constructing Regular Polygons Inscribed in Circles using Geometer's Sketchpad

Objective: SWBAT use Geometer's Sketchpad to construct a variety of regular polygons inscribed in circles.
Standards: HSG-CO.D.13
Subject(s): Math
60 minutes
1 Do Now - 5 minutes

As students walk in the room, I hand them a sheet with the names of 10 different polygons. Students write down the number of sides each polygon has. They are usually able to figure out 8 or 9 of the names based on prior knowledge. Some students may know that a dodecagon has 12 sides, but very few know that an icosagon has 20 sides. 

2 Mini-Lesson - 10 minutes

For this activity, students will be using Geometer’s Sketchpad to construct regular polygons inscribed in a circle. In a previous lesson, students have learned how to construct equilateral triangles, squares and regular hexagons using a compass and a straightedge. Although I refer back to this lesson, the procedure is different. This lesson is a bridge to the next unit, “Transformational Geometry. “

We begin the Mini-Lesson by reviewing the term “regular polygon.” A regular polygon is a polygon with congruent sides and angles. Students construct an equilateral triangle inscribed in a circle. They start by constructing a circle with center, A, and point B on the circle. Then students double click point A to mark it as the center of rotation. At this point, we discuss how to construct the regular polygon. I ask students, “What is the degree measure of each arc between two vertices on the circle of an inscribed equilateral triangle?” Students have difficulty with this at first, but then realize that there are three equal arcs with a measure of 120o  (360o divided by 3). The next step in the construction is to rotate point B 120o about point A and then repeat the rotation. Lastly, students connect the three points on the circle.

I usually have students measure the lengths of the sides of the triangle and the interior angles to show they are equal. We also review how to create a document with multiple pages. 

3 Activity - 25 minutes

Students repeat the process from the Mini-Lesson to construct a variety of regular polygons.  The steps are the same for each polygon. However, the degree of rotation is different. Students decide which regular polygon to construct and then figure out how many degrees to rotate a point on the circle in order to construct the polygon. After the construction, students name the polygon and show why it is a regular polygon. Each construction should be on a new page, with at least 4 constructions, not including the equilateral triangle.  As the students work, they complete a chart with information about the polygon and questions to answer and investigate.

While the students are working, I circulate around the room and help them with technical issues. Some students need help with calculations and measurements at first. 

After 20 minutes, I stop the students and we go over answers to questions 1 through 3 on the practice worksheet.

 Possible Answers:

  1. It is possible to construct a regular polygon by rotating a point on the circle 72o, but not by 52o because 360 is divisible by 72 and not by 52.
  2. When we constructed a regular hexagon using a compass and a straightedge, we used the radius of the circle as a guide for constructing the points on the circle. When we used Geometer’s Sketchpad, we divided 360 by 6 to find the measure of the arc between two vertices of the regular hexagon on the circle and rotated a point by that measure.
  3. Constructing a square using a compass and a straightedge involved constructing a perpendicular bisector of a diameter, which forms a 90o angle between each diameter. We used the intersections of these diameters and the circle as the vertices of the square. This is similar to rotating a point by 90o.
Constructing Inscribed Polygons using GSP Reflection
4 Summary - 5 minutes

Think-Pair-Share:   Why is it possible to construct a regular polygon by rotating a point around several times around a circle?  Students think for two minutes and then share their answer with the person sitting next to them. Then we have a short discussion of their responses.