‫بسم هللا الرحمن الرحيم‬
Geometrical Constructions
Bisect a Line (or Arc)
1. Draw two arcs of any radius greater than half-length of
the line with the centers at the ends of the line.
2. Join the intersection points of the arcs with a line.
3. Locate the midpoint.
Given
A
A
r1
B
r1
B
(not to scale)
Bisect an Angle
1. Draw an arc of any radius whose centers at the vertex.
2. Draw the arcs of any radius from the intersection
points between the previous arc and the lines.
3. Draw the line.
A
(not to scale)
Given
A
r2
r1
B
C
r2
C
B
Dividing a Line into a Number of Equal
Parts
To draw the line perpendicular to a
given line from a point not on the line
Adjacent-sides method
+
C
To draw the line perpendicular to a
given line from a point not on the line
Adjacent-sides method
+
C
Repeat
To draw the line perpendicular to a
given line from a point not on the line
Using compass
+ C
r2
D
r2
A
r1
B
Repeat
Tangents- Construction
Straight Line Tangents to a Circle from an External point
Tangents- Construction
Common Parallel Straight Line Tangents to Two Circles of
Radius ‘R’ and ‘r’
r
R
Tangents- Construction
Common Cross Straight Line Tangents to Two Circles of Radius
‘R’ and ‘r’
r
R
FILLET AND ROUND
Round
Sharp corner
Fillet
Round
Tangents- Construction
Circular Tangent of Radius ‘R’ Between a Point to a Straight
Line
R
R
R
R
Tangents- Construction
Circular Tangent of Radius R Between Two Straight Lines at
an Angle
R
R
R
R
R
To draw an arc of given radius tangent
to two lines
Given
arc radius r
T.P.1
T.P.2
Tangents- Construction
Internal Circular Tangent of Radius ‘R’ Between a Straight
Line and a Circle of Radius ‘r’
R+r
R
R
R
R
R
R
r
Tangents- Construction
External Circular Tangent of Radius ‘R’ Between a Straight
Line and a Circle of Radius ‘r’
R-r
R-r
R
R
R
R
R
R
r
FILLET AND ROUND
To draw the arc, we must find the location of the center of that arc.
How do we find the center of the arc?
Draw an arc of given radius tangent
to two perpendicular lines
Given
arc radius r
r
r
Draw an arc of given radius tangent
to two perpendicular lines
Given arc radius r
center of the arc
Starting point
Ending point
Construct an Arc Tangent to a Line
and an Arc
 Given line AB and arc CD.
 Strike arcs R1 (given radius).
 Draw construction arc parallel to
given arc, with center O.
 Draw construction line parallel to
given line AB.
 From intersection E, draw EO to get
tangent point T1, and drop
perpendicular to given line to get
point of tangency T2 .
 Draw tangent arc R from T1
to T2 with center E.
O
C
E
T1
R1
A
B
D
T2
When circle tangent to other circle
Tangent point
R1
C1
R2
C2
The center of two circles and tangent point lie on the same
straight line !!!
Draw a circle tangent to two circles I
Given
Example
C
+
+
+
C1
C2
Draw a circle tangent to two circles I
Given
Two circles and the radius of the third circle = R
+
C1
+
C2
Draw a circle tangent to two circles I
Given
Two circles and the radius of the third circle = R
center of the arc
R + R1
R + R2
C
R
R2
R1
+
C1
+
C2
Repeat
When circle tangent to other circle
Tangent point
R1
R2
C1
C2
The center of two circles and tangent point must lie on the
same straight line !!!
Draw a circle tangent to two circles II
Given
+
+
C1
C2
C+
Example
Draw a circle tangent to two circles II
Given
Two circles and the radius of the third circle = R
+
C1
+
C2
Draw a circle tangent to two circles II
Given
Two circles and the radius of the third circle = R
R
R2
R1
+
+
C2
C1
R – R1
C
R – R2
Repeat
Draw a circle tangent to two circles III
Given
Two circles and the radius of the third circle = R
R2
R1
+ C2
C1 +
R – R1
C
R + R2
Tangents- CW 1
85
85
R=18+22=40
R=44+22=66
END
Construct a Hexagon
given distance Across Flats (Circumscribed)
 Given distance across the
flats of a hexagon, draw
centerlines and a circle
with a diameter equal to
the distance across flats
 With parallel edge and
30° – 60 ° triangle, draw
the tangents
Construct a Hexagon
given distance Across Corners (Inscribed)
 Given distance AB across the corners, draw a circle
with AB as the diameter
 With A and B as centers and
the same radius, draw arcs
to intersect the circle at
points C, D, E, and F
 Connect the points to
complete the hexagon
C
D
A
B
F
E
Construct an Octagon
Across Flats (Circumscribed)
 Given the distance across the flats,
draw centerlines and a circle with a
diameter equal to the distance across
flats
 With a parallel edge and 45
triangle, draw lines tangent to
the circle in the order shown to
complete the octagon
1
5
7
3
4
8
6
2
Construct an Octagon
Across Corners (Inscribed)
 Given the distance across the
corners, draw centerlines AB
and CD and a circle with a
diameter equal to the distance
across corners
 With the T-square and 45°
triangle, draw diagonals EF
and GH
C
G
E
B
A
H
F
 Connect the points to
complete the octagon
D
General Method to Draw any Polygon
Note:
See Page (29) of Your Textbook
Draw an Approximate Ellipse
Given
Major and minor axes
Draw an approximate ellipse
Given
Major and minor axes
J
E
C
F
A
G
O
K
B
D
H
Repeat
Note:(Ellipse Drawing)
Also see Figure(3.24), page(43)
from textbook
END
‫طريقة رسم شكل سداسي‬
‫‪b‬‬
‫‪a‬‬
‫طريقة رسم شكل خماسي‬
‫‪b‬‬
‫‪a‬‬
‫طريقة عامة لرسم اى مضلع‬
‫‪e7‬‬
‫‪f7‬‬
‫‪d7‬‬
‫‪d5‬‬
‫‪c5‬‬
‫‪c4‬‬
‫‪c7‬‬
‫‪b‬‬
‫‪e5‬‬
‫‪7‬‬
‫‪6‬‬
‫‪5‬‬
‫‪4‬‬
‫‪g7‬‬
‫‪a‬‬