Introduction Construction methods can also be used to construct figures in a circle. One figure that...

IntroductionConstruction methods can also be used to construct figures in a circle. One figure that can be inscribed in a circle is a hexagon. Hexagons are polygons with six sides.

1.3.3: Constructing Regular Hexagons Inscribed in Circles

Key Concepts• Regular hexagons have six equal sides and six

angles, each measuring 120˚.

• The process for inscribing a regular hexagon in a circle is similar to that of inscribing equilateral triangles and squares in a circle.

• The construction of a regular hexagon is the result of the construction of two equilateral triangles inscribed in a circle.

Key Concepts, continued

Method 1: Constructing a Regular Hexagon Inscribed in a Circle Using a Compass

1. To construct a regular hexagon inscribed in a circle, first mark the location of the center point of the circle. Label the point X.

2. Construct a circle with the sharp point of the compass on the center point.

3. Label a point on the circle point A.4. Use a straightedge to connect point A and point X.

Extend the line through the circle, creating the diameter of the circle. Label the second point of intersection D.

(continued)

5. Without changing the compass setting, put the sharp point of the compass on A. Draw an arc to intersect the circle at two points. Label the points B and F.

6. Put the sharp point of the compass on D. Without changing the compass setting, draw an arc to intersect the circle at two points. Label the points C and E.

7. Use a straightedge to connect points A and B, B and C, C and D, D and E, E and F, and F and A.

Do not erase any of your markings.Hexagon ABCDEF is regular and is inscribed in circle X.

Key Concepts, continued• A second method “steps out” each of the vertices.

• Once a circle is constructed, it is possible to divide the circle into six equal parts.

• Do this by choosing a starting point on the circle and moving the compass around the circle, making marks equal to the length of the radius.

• Connecting every point of intersection results in a regular hexagon.

Method 2: Constructing a Regular Hexagon Inscribed in a Circle Using a Compass

1. To construct a regular hexagon inscribed in a circle, first mark the location of the center point of the circle. Label the point X.

2. Construct a circle with the sharp point of the compass on the center point.

3. Label a point on the circle point A.4. Without changing the compass setting, put the sharp

point of the compass on A. Draw an arc to intersect the circle at one point. Label the point of intersection B.

(continued)

5. Put the sharp point of the compass on point B. Without changing the compass setting, draw an arc to intersect the circle at one point. Label the point of intersection C.

6. Continue around the circle, labeling points D, E, and F. Be sure not to change the compass setting.

7. Use a straightedge to connect points A and B, B and C, C and D, D and E, E and F, and F and A.

Do not erase any of your markings.Hexagon ABCDEF is regular and is inscribed in circle X.

Common Errors/Misconceptions• inappropriately changing the compass setting

• attempting to measure lengths and angles with rulers and protractors

• not creating large enough arcs to find the points of intersection

• not extending segments long enough to find the vertices of the hexagon

Guided Practice

Example 1Construct regular hexagon ABCDEF inscribed in circle O using Method 1.

Guided Practice: Example 1, continued

1. Construct circle O.

Mark the location of the center point of the circle, and label the point O. Construct a circle with the sharp point of the compass on thecenter point.

2. Label a point on the circle point A.

3. Construct the diameter of the circle.Use a straightedge to connect point A and the center point, O. Extend the line through the circle, creating the diameter of the circle. Label the second point of intersection D.

4. Locate two vertices on either side of point A.Without changing the compass setting, put the sharp point of the compass on point A. Draw an arc to intersect the circle at two points.Label the points B and F.

5. Locate two vertices on either side of point D.Without changing the compass setting, put the sharp point of thecompass on point D. Draw an arc to intersect the circle at two points.Label the points C and E.

6. Construct the sides of the hexagon.Use a straightedge to connect A and B, B and C,C and D, D and E, E and F, and F and A, as shown on the next slide. Do not erase any of your markings.

Hexagon ABCDEF is a regular hexagon inscribed in circle O.

Guided Practice

Example 2Construct regular hexagon ABCDEF inscribed in circle O using Method 2.

1. Construct circle O.

Mark the location of the center point of the circle, and label the point O. Construct a circle with the sharp point of the compass on thecenter point.

2. Label a point on the circle point A.

3. Locate the remaining vertices.Without changing thecompass setting, put the sharp point of the compass on A. Draw an arc to intersect the circle at one point. Label the point of intersection B.

Guided Practice: Example 2, continuedPut the sharp point of the compass on point B. Without changing the compass setting, draw an arc to intersect the circle at one point. Label the point of intersection C.

Guided Practice: Example 2, continuedContinue around the circle, labeling points D, E, and F. Be sure not to change the compass setting.

4. Construct the sides of the hexagon.Use a straightedge to connect A and B, B and C, C and D, D and E, E and F, and F and A, as shown on the next slide. Do not erase any of your markings.

Hexagon ABCDEF is a regular hexagon inscribed in circle O.

Introduction Construction methods can also be used to construct figures in a circle. One figure that...

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