Constructing Inscribed Regular Triangles, Quadrilaterals and Hexagons

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Objective

SWBAT construct regular triangles, quadrilaterals and hexagons inscribed in circles using a compass and a straightedge.

Big Idea

Students will use concepts from previous lessons to construct equilateral triangles, squares and regular hexagons inscribed in a circle.

Do Now

6 minutes

As students walk into the classroom, I hand them a compass and a straightedge. They draw two segments of different sizes and construct the perpendicular bisector.

As the students are working, I walk around the room to ensure they are able to bisect the segments. If students are still having difficulties, I pair them with other students who can help.

Mini-Lesson

14 minutes

The first part of the Mini-Lesson involves constructing a regular hexagon that is inscribed in a circle. We discuss the terms "regular" and "inscribed." It is also helpful to talk about what is means for a point to be "on the circle." I think show the students how to construct a regular hexagon inscribed in a circle. 

Constructing an equilateral triangle inscribed in a circle involves most of the same steps as constructing the hexagon. The only difference is in the last step. Instead of connecting consecutive points on the circle, students connect every other point. This creates the equilateral triangle.  

*See the video, "Constructing a Regular Hexagon and an Equilateral Triangle Inscribed in a Circle," for a demonstration of the constructions.

Constructing a square inscribed in a circle involves constructing the perpendicular bisector of a diameter. Students begin by marking a point for the center of a circle and constructing a circle. Then students draw a diameter and mark its endpoints on the circle. I usually discuss the definition and/or properties of a diameter. Students then construct the perpendicular bisector of the diameter and mark off the points where the bisector intersect the circle. The last step is to connect consecutive points. The four points on the circle become the vertices of the square. Students may not realize they have constructed a square and may refer to their figure as a "diamond." I discuss the properties of a square, including congruent, perpendicular diagonals, and we verify these properties in the construction.

Practice

15 minutes

Students practice constructing a regular hexagon, an equilateral triangle and a square inscribed in a circle. As students practice, I circulate around the room. If I notice common misunderstandings, I stop the students and show them the steps again.

After students perform their constructions, they calculate the degree measures of the arcs formed by consecutive vertices of the inscribed figures. Students may have difficulty at first, but after they figure out the arc lengths in the first construction, the other calculations are easier.

Students may need a reminder about the difference between the length of an arc and its degree measure. The degree measures of the arc will be adressed again in a later lesson.

At the end of the practice section, I show some student examples and we discuss any misconceptions. 

Summary

10 minutes

Ask:  Why is it possible to construct an equilateral triangle using the construction for a regular hexagon? Are there any other inscribed regular polygons that can be constructed using the constructions we learned today?

Students can construct an inscribed regular octagon by bisecting two consecutive sides of a square. They can construct a dodecagon by bisecting angles formed by two consecutive diagonals in the hexagon.