Introduction
The ability to copy and bisect angles and segments, as
well as construct perpendicular and parallel lines, allows
you to construct a variety of geometric figures, including
triangles, squares, and hexagons. There are many ways
to construct these figures and others. Sometimes the
best way to learn how to construct a figure is to try on
your own. You will likely discover different ways to
construct the same figure and a way that is easiest for
you. In this lesson, you will learn two methods for
constructing a triangle within a circle.
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5.4.1: Constructing Equilateral Triangles Inscribed in Circles
Key Concepts
Triangles
• A triangle is a polygon with three sides and three
angles.
• There are many types of triangles that can be
constructed.
• Triangles are classified based on their angle measure
and the measure of their sides.
• Equilateral triangles are triangles with all three sides
equal in length.
• The measure of each angle of an equilateral triangle
is 60˚.
5.4.1: Constructing Equilateral Triangles Inscribed in Circles
2
Key Concepts, continued
Circles
• A circle is the set of all points that are equidistant from a
reference point, the center.
• The set of points forms a two-dimensional curve
that is 360˚.
• Circles are named by their center. For example, if a
circle has a center point, G, the circle is named circle G.
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5.4.1: Constructing Equilateral Triangles Inscribed in Circles
Key Concepts, continued
• The diameter of a circle is a straight line that goes
through the center of a circle and connects two points
on the circle. It is twice the radius.
• The radius of a circle is a line segment that runs from
the center of a circle to a point on the circle.
• The radius of a circle is one-half the length of the
diameter.
• There are 360˚ in every circle.
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5.4.1: Constructing Equilateral Triangles Inscribed in Circles
Key Concepts, continued
Inscribing Figures
• To inscribe means to draw a figure within another figure
so that every vertex of the enclosed figure touches the
outside figure.
• A figure inscribed within a circle is a figure drawn within
a circle so that every vertex of the figure touches the
circle.
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5.4.1: Constructing Equilateral Triangles Inscribed in Circles
Key Concepts, continued
• It is possible to inscribe a triangle within a circle. Like
with all constructions, the only tools used to inscribe a
figure are a straightedge and a compass, patty paper
and a straightedge, reflective tools and a straightedge,
or technology.
• This lesson will focus on constructions with a compass
and a straightedge.
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5.4.1: Constructing Equilateral Triangles Inscribed in Circles
Key Concepts, continued
Method 1: Constructing an Equilateral Triangle Inscribed
in a Circle Using a Compass
1. To construct an equilateral triangle 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 and draw an arc to intersect the
circle at two points. Label the points B and C.
5. Use a straightedge to construct BC.
(continued)
5.4.1: Constructing Equilateral Triangles Inscribed in Circles
7
Key Concepts, continued
6. Put the sharp point of the compass on point B. Open the
compass until it extends to the length of BC. Draw another
arc that intersects the circle. Label the point D.
7. Use a straightedge to construct BD and CD.
Do not erase any of your markings.
Triangle BCD is an equilateral triangle inscribed in circle X.
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5.4.1: Constructing Equilateral Triangles 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 6 equal parts.
• Do this by choosing a starting point on the circle and
moving the compass around the circle, making marks
that are the length of the radius apart from one another.
• Connecting every other point of intersection results in an
equilateral triangle.
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5.4.1: Constructing Equilateral Triangles Inscribed in Circles
Key Concepts, continued
Method 2: Constructing an Equilateral Triangle Inscribed
in a Circle Using a Compass
1. To construct an equilateral triangle 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 and draw an arc to intersect the
circle at one point. Label the point of intersection B.
(continued)
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5.4.1: Constructing Equilateral Triangles Inscribed in Circles
Key Concepts, continued
5. Put the sharp point of the compass on point B and 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 A and C, C and E, and E
and A.
Do not erase any of your markings.
Triangle ACE is an equilateral triangle inscribed in circle X.
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5.4.1: Constructing Equilateral Triangles 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
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5.4.1: Constructing Equilateral Triangles Inscribed in Circles
Guided Practice
Example 3
Construct equilateral triangle JKL inscribed in circle P
using Method 1. Use the length of HP as the radius for
circle P.
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5.4.1: Constructing Equilateral Triangles Inscribed in Circles
Guided Practice: Example 3, continued
1. Construct circle P.
Mark the location of the
center point of the circle,
and label the point P.
Set the opening of the
compass equal to the
length of HP. Then, put
the sharp point of the
compass on point P and
construct a circle. Label
a point on the circle
point G.
5.4.1: Constructing Equilateral Triangles Inscribed in Circles
14
Guided Practice: Example 3, continued
2. Locate vertices J and K of the equilateral
triangle.
Without changing the compass setting, put the sharp
point of the compass on G. Draw an arc to intersect
the circle at two points. Label the points J and K, as
shown on the next slide.
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5.4.1: Constructing Equilateral Triangles Inscribed in Circles
Guided Practice: Example 3, continued
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5.4.1: Constructing Equilateral Triangles Inscribed in Circles
Guided Practice: Example 3, continued
3. Locate the third vertex of the equilateral
triangle.
Put the sharp point of the compass on point J. Open
the compass until it extends to the length of JK .
Draw another arc that intersects the circle, and label
the point L, as shown on the next slide.
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5.4.1: Constructing Equilateral Triangles Inscribed in Circles
Guided Practice: Example 3, continued
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5.4.1: Constructing Equilateral Triangles Inscribed in Circles
Guided Practice: Example 3, continued
4. Construct the sides of the triangle.
Use a straightedge to connect J and K, K and L, and
L and J, as shown on the next slide. Do not erase
any of your markings.
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5.4.1: Constructing Equilateral Triangles Inscribed in Circles
Guided Practice: Example 3, continued
Triangle JKL is an equilateral triangle
inscribed in circle P with the given radius.
✔
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5.4.1: Constructing Equilateral Triangles Inscribed in Circles
Guided Practice: Example 3, continued
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5.4.1: Constructing Equilateral Triangles Inscribed in Circles
Guided Practice
Example 4
Construct equilateral triangle JLN inscribed in circle P
using Method 2. Use the length of HP as the radius for
circle P.
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5.4.1: Constructing Equilateral Triangles Inscribed in Circles
Guided Practice: Example 4, continued
1. Construct circle P.
Mark the location of the
center point of the circle,
and label the point P.
Set the opening of the
compass equal to the
length of HP. Then, put
the sharp point of the
compass on point P and
construct a circle. Label
a point on the circle
point G.
5.4.1: Constructing Equilateral Triangles Inscribed in Circles
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Guided Practice: Example 4, continued
2. Locate vertex J.
Without changing the
compass setting, put the
sharp point of the
compass on G. Draw
an arc to intersect
the circle at one point.
Label the point of
intersection J.
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5.4.1: Constructing Equilateral Triangles Inscribed in Circles
Guided Practice: Example 4, continued
Put the sharp point of the compass on point J.
Without changing the compass setting, draw an arc
to intersect the circle at one point. Label the point of
intersection K.
Continue around the circle, labeling points L, M, and
N, as shown on the next slide. Be sure not to change
the compass setting.
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5.4.1: Constructing Equilateral Triangles Inscribed in Circles
Guided Practice: Example 4, continued
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5.4.1: Constructing Equilateral Triangles Inscribed in Circles
Guided Practice: Example 4, continued
3. Construct the sides of the triangle.
Use a straightedge to connect J and L, L and N, and
J and N, as shown on the next slide. Do not erase
any of your markings.
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5.4.1: Constructing Equilateral Triangles Inscribed in Circles
Guided Practice: Example 4, continued
Triangle JLN is an equilateral triangle
inscribed in circle P.
✔
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5.4.1: Constructing Equilateral Triangles Inscribed in Circles
Guided Practice: Example 4, continued
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5.4.1: Constructing Equilateral Triangles Inscribed in Circles