Geometric Construction Notes 2
Table of Contents
Construct a 45 Degree
Line
Construct an Inscribed
Square From a Side
Construct a Circumscribed
Square from a Side
Inscribed Hexagon
Inscribed Octagon from
Square
Divide Line Into Equal
Parts
Construct an Arc Tangent
to Two Lines
Construct an Arc Tangent
to an Arc and a Line
Construct an Arc Tangent
to Two Arcs
Construct a 45 degree line with the
Compass
Begin with a line
1. Bisect the line
Construct a 45 degree line with the
Compass
2. Measure the length of ½ of
the line from the bisector
to the ep.
3. Without adjusting the
compass radius, draw an
arc intersecting the
bisector
Construct a 45 degree line with the
Compass
4. Connect the ep of the line
to the intersection of the
arc and bisector
Construct a 45 degree line with the
Compass- Solution
Construct an Inscribed Square from
a Side
Begin with a Side
1. Draw intersecting
centerlines (CL)
from the end points
of the given
diagonal – use 45
degree angle
construction.
Construct an Inscribed Square from
a Diagonal Side
2. Set the compass point on the intersection of
the cls and adjust the radius of the compass
to one of the eps of the diagonal.
3. Use the radius to
draw a circle. Both
eps of the line
should touch the
perimeter of the
circle
Construct an Inscribed Square from
a Diagonal Side
4. Connect the intersections of the cls and the
circle perimeter to form the missing sides.
Construct a Circumscribed Square
from a Side
Begin with a Side
1. Draw intersecting
lines from the end
points of the given
diagonal – use 45
degree angle
construction.
Construct a Circumscribed Square
from a Side
2. Bisect the given line to find the radius of the
circle. Your construction may overlap other
constructions
Note: Extend the bisector
on both sides of the cp
Construct a Circumscribed Square
from a Side
3. Draw a line
perpendicular
to the
bisector at
the cp using
the steps for
drawing a
perpendicular
from a point
on a line .
Construct a Circumscribed Square
from a Side
4. Measure the radius of
the circle by setting
the point of the
compass to the
intersection of the
two cls, the center
point (cp), and the
lead to the
intersection of the
bisect and given side
5. Draw a circle that
intersects the
diagonals and cls.
Construct a Circumscribed Square
from a Side
6. Draw lines from the
ep of the given line
tangent to the circle
at the intersection
of the cls and circle
perimeter to the
diagonal
Construct a Circumscribed Square
from a Side
7. Draw a fourth side
by connecting the
eps of the two
sides drawn
passing through
the tangent point
of the circle
Construct a Circumscribed Square
from a Side- Solution
Construct an Inscribed Hexagon to
a Given Circle
Begin with a circle and cp
1. Set the compass point
at any point along the
circle perimeter and
measure the circle
radius with the
compass
Construct an Inscribed Hexagon to
a Given Circle
2. Without adjusting the compass radius or moving the
point draw and arc intersecting the perimeter of the
circle.
3. Without adjusting the compass radius move the
compass point to the intersection of the arc and
circle and draw a second arc intersecting the circle
Construct an Inscribed Hexagon to
a Given Circle
4. Repeat step 3 until you
have 5 arcs
intersecting the circle
perimeter. These
along with the original
point are the vertices
of the hexagon.
Construct an Inscribed Hexagon to
a Given Circle
5. Connect the vertices
with a straight edge to
create the sides
Construct an Inscribed Octagon to
a Given Square
Begin with a square
1. Find the cp of the square by
connecting opposite
corners of the square
Construct an Inscribed Octagon to
a Given Square
3. Use the compass to
measure the distance
from one vertices of the
square and the cp
4. Without adjusting the
compass radius draw arcs
from each vertices
intersecting the adjacent
sides
Construct an Inscribed Octagon to
a Given Square
5. Connect intersections of
the arcs and perimeter of
the square
Construct an Inscribed Octagon to
a Given Square- Solution
Divide a Line Into Equal Parts
Begin with a line
1. Draw a diagonal line
from one ep of the
line
2. Using the compass draw arcs
intersecting the angle to
create equal spacing.
Note: The number of spaces is determined by the number
of divisions
Divide a Line Into Equal Parts
3. Using the compass
measure the distance
between the ep of the
line and last
intersection along the
diagonal.
4. Without adjusting the compass radius
draw an arc from the opposite ep.
Divide a Line Into Equal Parts
5. Using the compass
measure the length of
the diagonal to the last
intersection.
6. Without adjusting the compass radius
draw an arc from the opposite ep.
that intersects the other arc
Divide a Line Into Equal Parts
7. Use a straightedge to connect the ep of the
line and the intersection. This line is parallel
to the diagonal.
Divide a Line Into Equal Parts
8. Using the same radius as the first set of
intersecting arcs draw an identical set.
Divide a Line Into Equal Parts
9. Connect the intersections
of the arcs and diagonals
with parallel lines.
The parallel lines divide the
given line into equal parts
where they intersect the
given line.
Divide a Line Into Equal PartsSolution
Construct an Arc Tangent to Two
Lines
Begin with any angle or pair of
non-parallel lines
1. Use the steps to
draw a parallel
line through a
point. The point
must be the
same distance
away from the
line as the radius
of the arc
Note: You may use a scale to locate
the point
Construct an Arc Tangent to Two
Lines
2. Use the compass to measure from
the intersection of the lines, the cp,
to the intersection of one of the
lines and a given line, the radius.
3. Draw the arc between
both lines.
Construct an Arc Tangent to Two
Lines- Solution
Construct an Arc Tangent to an Arc
and a Line
Begin with the cp of an arc and a given line
1. Use the steps to draw
a parallel line to the
given line the same
distance away as the
radius of the tangent
arc
Note: You may use a scale to locate
the point
Construct an Arc Tangent to an Arc
and a Line
2. 2. Draw the circle or
arc to the given
radius or diameter
Construct an Arc Tangent to an Arc
and a Line
3. Draw a concentric arc
with a radius equal
to the radius of the
given arc or circle
plus the radius of the
tangent arc
Construct an Arc Tangent to an Arc
and a Line
4.Using the steps to
draw a perpendicular
through a point on a
line draw a center
line at the
intersection of the
arc and line
Construct an Arc Tangent to an Arc
and a Line
5.Using the steps to
draw a perpendicular
through a point on a
line draw a center
line at the
intersection of the
arc and line
Construct an Arc Tangent to an Arc
and a Line
6. Set the compass point
to the intersection of
the line and arc, the
cp, and measure the
radius to the
intersection of the
given line and
perpendicular
7. Draw the arc tangent to the
circle and line
Construct an Arc Tangent to an Arc
and a Line- Solution
Construct an Arc Tangent to Two
Arcs
Begin with the cps of two arcs
1. Draw the circles or arcs to the given
radii or diameters
Construct an Arc Tangent to Two
Arcs
2. Draw intersecting concentric arcs with a radii equal to
the radius of each of the given arcs or circles plus the
radius of the tangent arc
Construct an Arc Tangent to Two
Arcs
3. Draw a line from the cp of one of the circles or arcs and
the intersection of the arcs
Construct an Arc Tangent to Two
Arcs
4. Place the compass point on the intersection of the arcs
and the intersection of the circle and line.
5. Draw the tangent arc.
Construct an Arc Tangent to Two
Arcs- Solution