# MA.912.GR.5.4

Construct a regular polygon inscribed in a circle. Regular polygons are limited to triangles, quadrilaterals and hexagons.

### Clarifications

Clarification 1: When given a circle, the center must be provided.

Clarification 2: Instruction includes using compass and straightedge, string, reflective devices, paper folding or dynamic geometric software.

General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Geometric Reasoning
Status: State Board Approved

## Benchmark Instructional Guide

### Terms from the K-12 Glossary

• Inscribed Polygon in a Circle

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grade 8, students learned about regular polygons and their interior angles. In Geometry, students inscribe such polygons in circles using compass and straightedge.
• Instruction includes the use of manipulatives, tools and geometric software. Allowing students to explore constructions with dynamic software reinforces why the constructions work.
• Instruction includes the student understanding that in a geometric construction, one does not use the markings on a ruler or on a protractor to construct inscribed and circumscribed circles of a triangle. Students should realize that there are limitations on precision that are inherent in the markings on rulers or protractors.
• It is important to build the understanding that formal constructions are valid when the lengths of segments or measures of angles are not known, or have values that do not appear on a ruler or protractor, including irrational values.
• Instruction includes the connection to logical reasoning and visual proofs when verifying that a construction works.
• To construct a square inscribed in a circle given the center of the circle, the procedure can start by drawing one of the diameters of the circle. This diameter will be one of the diagonals of the square. Then, students can construct the perpendicular bisector of the drawn diameter. The perpendicular bisector will intersect the circle in two points resulting in another diameter of the circle that is congruent and perpendicular to the drawn diameter. Students should realize that diagonals of squares have the same length and are perpendicular bisectors of one another. Therefore, the two diameters are the diagonals of the square to be constructed in the circle. To construct this square, draw the sides by connecting each of the endpoints of the diagonals.
• The construction of a regular hexagon inscribed in a circle (given the center) can start by choosing one point, P on the circle, then, with the compass setting equal to the radius of the circle, students can draw an arc intersecting the circle at the point Q. Students should realize that the two points on the circle and the center of the circle are the vertices of an equilateral triangle. Students can then move the compass to the point Q and draw another arc intersecting the circle at the point R. Students can repeat this process as they move around the circle until the arrive back at the point P. Students should realize that as they are repeating this process around the circle, they are forming six equilateral triangles. These triangles compose the regular hexagon that is inscribed in the circle.

• The construction of an equilateral triangle can begin by constructing a regular hexagon, as described above. Once the six equilateral triangles have been formed, to construct an equilateral triangle in the circle, students can join three pairs of alternating vertices (as shown below).

• Instruction includes the connection to the sum of the measures of the interior angles of a regular polygon to the construction of equilateral triangles and hexagons inscribed in a circle.
• Enrichment of this benchmark includes constructing other regular polygons, such as octagons and dodecagons, using angle bisection.
• For expectations of this benchmark, constructions should be reasonably accurate and the emphasis is to make connections between the construction steps and the definitions, properties and theorems supporting them.
• While going over the steps of geometric constructions, ensure that students develop vocabulary to describe the steps precisely. (MTR.4.1)
• Problem types include identifying the next step of a construction, a missing step in a construction or the order of the steps in a construction.

### Common Misconceptions or Errors

• Students may think or determine that when constructing a regular hexagon, they will not have all equal side lengths. To help address this, reiterate the importance of precision and accuracy when using a compass.

• Circle A is provided below.

• Part A. What do you know about all the points on circle A in relation to point A?
• Part B. Draw a point J on circle A. Open the compass to the length of the resulting radius of circle A.
• Part C. With the compass point on point J, arc on both sides of J making sure to intersect circle A. Label the points of intersection as J and M.
• Part D. Classify triangle AKJ by sides.
• Part E. What is the measure of angle KJM?
• Part F. What is the measure of an interior angle of a regular hexagon?
• Part G. What process could be used to continue constructing a regular hexagon that is inscribed in circle A?
• Part H. How could the construction of an inscribed regular hexagon be used to construct an inscribed equilateral triangle?

Instructional Task 2 (MTR.3.1, MTR.4.1, MTR.5.1)
• Part A. Construct a regular hexagon inscribed in a circle.
• Part B. Prove that the constructed hexagon is a regular hexagon.
• Part C. Prove that if three pairs of alternating vertices of the hexagon from Part A are joined, it creates an equilateral triangle.
• Part D. Compare your proof from Part C with a partner.

### Instructional Items

Instructional Item 1
• Describe the steps to construct a square inscribed in circle J.

*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.

## Related Courses

This benchmark is part of these courses.
1200400: Foundational Skills in Mathematics 9-12 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1206320: Geometry Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.

## Related Resources

Vetted resources educators can use to teach the concepts and skills in this benchmark.

## Formative Assessments

Regular Hexagon in a Circle:

Students are asked to construct a regular hexagon inscribed in a circle.

Type: Formative Assessment

Equilateral Triangle in a Circle:

Students are asked to construct an equilateral triangle inscribed in a circle.

Type: Formative Assessment

Construct the Center of a Circle:

Students are asked to construct the center of a circle.

Type: Formative Assessment

Square in a Circle:

Students are asked to construct a square inscribed in a circle.

Type: Formative Assessment

## Lesson Plans

Construction of Inscribed Regular Hexagon:

A GeoGebra lesson for students to become familiar with computer based construction tools. Students work together to construct a regular hexagon inscribed in a circle using rotations. Directions for both a beginner and advanced approach are provided.

Type: Lesson Plan

Inscribe Those Rims:

This lesson will engage students with an interactive and interesting way to learn how to inscribe polygons in circles.

Type: Lesson Plan

Construction Junction:

Students will learn how to construct an equilateral triangle and a regular hexagon inscribed in a circle using a compass and a straightedge.

Type: Lesson Plan

Inscribe it:

This activity allows students to practice the construction process inscribing a regular hexagon and an equilateral triangle in a circle using GeoGebra software.

Type: Lesson Plan

Construct Regular Polygons Inside Circles:

Students will be able to demonstrate that they can construct, using the central angle method, an equilateral triangle, a square, and a regular hexagon, inscribed inside a circle, using a compass, straightedge, and protractor. They will use worksheets to master the construction of each polygon, one inside each of three different circles. As an extension to this lesson, if computers with GeoGebra are available, the students should be able to perform these constructions on computers as well.

Type: Lesson Plan

Construct This:

In this lesson, students will construct a square inscribed in a circle using the properties of a square and determine if there is more than one way to complete the construction.

Type: Lesson Plan

Determination of the Optimal Point:

Students will use dynamic geometry software to determine the optimal location for a facility under a variety of scenarios. The experiments will suggest a relation between the optimal point and a common concept in geometry; in some cases, there will be a connection to a statistical concept. Algebra can be used to verify some of the conjectures.

Type: Lesson Plan

Paper Plate Origami:

A hands-on activity where students construct inscribed regular polygons in a circle using models. Through guided questions, students will discover how to divide a model (paper plate) into 3, 4, and 6 parts. Using folding, a straightedge, and a compass, they will construct an equilateral triangle, a square, and a regular hexagon in their circles.

Type: Lesson Plan

Crafty Circumference Challenge:

Students learn about geometric construction tools and how to use them. Students will partition the circumference of a circle into three, four, and six congruent arcs which determine the vertices of regular polygons inscribed in the circle. An optional project is included where students identify, find, and use recycled, repurposed, or reclaimed objects to create "crafty" construction tools.

Type: Lesson Plan

St. Pi Day construction with a compass & ruler:

St. Pi Day construction with compass

This activity uses a compass and straight-edge(ruler) to construct a design. The design is then used to complete a worksheet involving perimeter, circumference, area and dimensional changes which affect the scale factor ratio.

Type: Lesson Plan

Concurrent Points Are Optimal:

Students will begin with a review of methods of construction of perpendicular bisectors and angle bisectors for the sides of triangles. Included in the review will be a careful discussion of the proofs that the constructions actually produce the lines that were intended.

Next, students will investigate why the perpendicular bisectors and angle bisector are concurrent, that is, all three meet at a single meet.

A more modern point of currency is the Fermat-Torricelli point (F-T). The students will construct (F-T) in GeoGebra and investigate limitations of its existence for various types of triangles.

Then a set of scenarios will be provided, including some one-dimensional and two-dimensional situations. Students will use GeoGebra to develop conjectures regarding whether a point of concurrency provides the solution for the indicated situation, and which one.

A physical model for the F-T will be indicated. The teacher may demonstrate this model but that requires three strings, three weights, and a base that has holes. A recommended base is a piece of pegboard (perhaps 2 feet by 3 feet), the weights could be fishing weights of about 3 oz., the string could be fishing line; placing flexible pieces of drinking straws in the holes will improve the performance.

The combination of geometry theorems, dynamic geometry software, a variety of contexts, and a physical analog can provide a rich experience for students.

Type: Lesson Plan

## Original Student Tutorials

A Square Peg in a Round Hole:

Learn how to construct an inscribed square in a circle and why certain constructions are used in this interactive tutorial.

Type: Original Student Tutorial

Designing with Hexagons:

Learn how to construct an inscribed regular hexagon and equilateral triangle in a circle in this interactive tutorial.

Type: Original Student Tutorial

Inscribing a hexagon in a circle:

This problem solving task challenges students to inscribe equilateral triangles and regular hexagons on a circle with a compass and straightedge.

## MFAS Formative Assessments

Construct the Center of a Circle:

Students are asked to construct the center of a circle.

Equilateral Triangle in a Circle:

Students are asked to construct an equilateral triangle inscribed in a circle.

Regular Hexagon in a Circle:

Students are asked to construct a regular hexagon inscribed in a circle.

Square in a Circle:

Students are asked to construct a square inscribed in a circle.

## Original Student Tutorials Mathematics - Grades 9-12

A Square Peg in a Round Hole:

Learn how to construct an inscribed square in a circle and why certain constructions are used in this interactive tutorial.

Designing with Hexagons:

Learn how to construct an inscribed regular hexagon and equilateral triangle in a circle in this interactive tutorial.

## Student Resources

Vetted resources students can use to learn the concepts and skills in this benchmark.

## Original Student Tutorials

A Square Peg in a Round Hole:

Learn how to construct an inscribed square in a circle and why certain constructions are used in this interactive tutorial.

Type: Original Student Tutorial

Designing with Hexagons:

Learn how to construct an inscribed regular hexagon and equilateral triangle in a circle in this interactive tutorial.

Type: Original Student Tutorial

Inscribing a hexagon in a circle:

This problem solving task challenges students to inscribe equilateral triangles and regular hexagons on a circle with a compass and straightedge.