Introduction
Construction 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
5.4.3: 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.
2
5.4.3: Constructing Regular Hexagons Inscribed in Circles
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)
3
5.4.3: Constructing Regular Hexagons Inscribed in Circles
Key Concepts, 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.
4
5.4.3: Constructing Regular Hexagons Inscribed in Circles
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.
5
5.4.3: Constructing Regular Hexagons Inscribed in Circles
Key Concepts, continued
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)
6
5.4.3: Constructing Regular Hexagons Inscribed in Circles
Key Concepts, 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.
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5.4.3: Constructing Regular Hexagons Inscribed in Circles
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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5.4.3: Constructing Regular Hexagons Inscribed in Circles
Guided Practice
Example 1
Construct regular hexagon ABCDEF inscribed in circle O
using Method 1.
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5.4.3: Constructing Regular Hexagons Inscribed in Circles
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 the
center point.
10
5.4.3: Constructing Regular Hexagons Inscribed in Circles
Guided Practice: Example 1, continued
2. Label a point on the circle point A.
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5.4.3: 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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5.4.3: 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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5.4.3: 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 the
compass on point D.
Draw an arc to intersect
the circle at two points.
Label the points C and E.
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5.4.3: 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.
15
5.4.3: Constructing Regular Hexagons Inscribed in Circles
Guided Practice: Example 1, continued
Hexagon ABCDEF is a regular hexagon inscribed in
circle O.
✔
16
5.4.3: Constructing Regular Hexagons Inscribed in Circles
Guided Practice: Example 1, continued
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5.4.3: Constructing Regular Hexagons Inscribed in Circles
Guided Practice
Example 2
Construct regular hexagon ABCDEF inscribed in circle O
using Method 2.
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5.4.3: Constructing Regular Hexagons Inscribed in Circles
Guided Practice: Example 2, 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 the
center point.
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5.4.3: Constructing Regular Hexagons Inscribed in Circles
Guided Practice: Example 2, continued
2. Label a point on the circle point A.
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5.4.3: Constructing Regular Hexagons Inscribed in Circles
Guided Practice: Example 2, continued
3. Locate the remaining vertices.
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.
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5.4.3: Constructing Regular Hexagons Inscribed in Circles
Guided Practice: Example 2, continued
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.
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5.4.3: Constructing Regular Hexagons Inscribed in Circles
Guided Practice: Example 2, continued
Continue around the
circle, labeling points
D, E, and F. Be sure
not to change the
compass setting.
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5.4.3: 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.
24
5.4.3: Constructing Regular Hexagons Inscribed in Circles
Guided Practice: Example 2, continued
Hexagon ABCDEF is a regular hexagon inscribed in
circle O.
✔
25
5.4.3: Constructing Regular Hexagons Inscribed in Circles
Guided Practice: Example 2, continued
26
5.4.3: Constructing Regular Hexagons Inscribed in Circles