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.

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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.

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1.3.3: Constructing Regular Hexagons Inscribed in Circles

Key Concepts, continued

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1.3.3: Constructing Regular Hexagons Inscribed in Circles

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)

Key Concepts, continued

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1.3.3: Constructing Regular Hexagons Inscribed in Circles

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.

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1.3.3: Constructing Regular Hexagons Inscribed in Circles

Key Concepts, continued

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1.3.3: Constructing Regular Hexagons Inscribed in Circles

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)

Key Concepts, continued

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1.3.3: Constructing Regular Hexagons Inscribed in Circles

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

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1.3.3: Constructing Regular Hexagons Inscribed in Circles

Guided Practice

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

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1.3.3: Constructing Regular Hexagons Inscribed in Circles

Guided Practice: Example 1, continued

1. Construct circle O.

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1.3.3: Constructing Regular Hexagons Inscribed in Circles

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.

Guided Practice: Example 1, continued

2. Label a point on the circle point A.

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1.3.3: Constructing Regular Hexagons Inscribed in Circles

Guided Practice: Example 1, continued

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.

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1.3.3: Constructing Regular Hexagons Inscribed in Circles

Guided Practice: Example 1, continued

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.

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1.3.3: Constructing Regular Hexagons Inscribed in Circles

Guided Practice: Example 1, continued

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.

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1.3.3: Constructing Regular Hexagons Inscribed in Circles

Guided Practice: Example 1, continued

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.

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1.3.3: Constructing Regular Hexagons Inscribed in Circles

Guided Practice: Example 1, continued

Hexagon ABCDEF is a regular hexagon inscribed in circle O.

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1.3.3: Constructing Regular Hexagons Inscribed in Circles

✔

Guided Practice: Example 1, continued

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1.3.3: Constructing Regular Hexagons Inscribed in Circles

Guided Practice

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

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1.3.3: Constructing Regular Hexagons Inscribed in Circles

Guided Practice: Example 2, continued

1. Construct circle O.

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1.3.3: Constructing Regular Hexagons Inscribed in Circles

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.

Guided Practice: Example 2, continued

2. Label a point on the circle point A.

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1.3.3: Constructing Regular Hexagons Inscribed in Circles

Guided Practice: Example 2, continued

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.

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1.3.3: Constructing Regular Hexagons Inscribed in Circles

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.

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1.3.3: Constructing Regular Hexagons Inscribed in Circles

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

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1.3.3: Constructing Regular Hexagons Inscribed in Circles

Guided Practice: Example 2, continued

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.

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1.3.3: Constructing Regular Hexagons Inscribed in Circles

Guided Practice: Example 2, continued

Hexagon ABCDEF is a regular hexagon inscribed in circle O.

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1.3.3: Constructing Regular Hexagons Inscribed in Circles

✔

Guided Practice: Example 2, continued

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1.3.3: Constructing Regular Hexagons Inscribed in Circles