Download Brief outline of types of construction:

Carpenter’s Geometric
Constructions
Kevin Mirus
Madison Area Technical College
Madison East High School Math Week
Monday, May 16, 2005
[email protected]
http://www.matcmadison.edu/is/as/math/kmirus/
Page 1 of 17
1. The Pythagorean Theorem: a2 + b2 = c2 (for right triangles)
Application: A triangle with sides of length 3, 4, and 5 (or multiples thereof)
automatically forms a right triangle. This can be used to check that an angle is
square, or to lay out a square angle.
2. Copying line lengths
Problem: Construct a line segment congruent to a given segment.
Given: Line segment
AB
Step 1: Choose any
point C on line l.
Step 2: Use a compass
to measure the distance
AB, then draw an arc
centered at C with
radius AB that intersects
l at point D.
Comments: This may seem overly simple, but it has a lot of practical uses.
Application: Create an equilateral triangle, which has 60 angles.
Problem: Construct an equilateral triangle.
Comments:
Application: You can use this to copy triangles, just by copying the lengths of
each of the three sides. You can also use this to copy any polygon, since you can
always divide it up into triangles.
Page 2 of 17
3. Constructing Perpendicular Lines
Application: To get the exact center of a distance without measuring, you can
construct its perpendicular bisector:
Problem: Construct the perpendicular bisector of a given line segment.
Step 1: Draw arcs centered Step 2: Draw segment PQ .
on A and B that have the
Label its intersection with
same radius greater than the
AB as point M.
distance to the midpoint.
Label the intersections P
and Q.
Comments: Constructing the perpendicular bisector gives you both the midpoint and a
right angle!
Given: Line segment AB
Problem: Construct a line perpendicular to a given line and through a given
point.
Starting with line segment AB and point C not on the line, strike an arc centered at C so
that the arc intersects AB at two points (E and F). Construct the perpendicular bisector
between EF, and it will pass through point C.
Comments:
Page 3 of 17
Application: This technique allows you to make squares and rectangles.
Application: To find the center of a circle, pick any two random chords, then
construct the perpendicular bisectors of the chords. The bisectors will intersect at the
center of the circle.
Problem: Locate the center of a given circle.
Comments:
Page 4 of 17
Application: Finding the center of an arch
Page 5 of 17
Bisecting angles
Problem: Construct the bisector of a given angle.
Given: Angle BAC
Step 1: Draw an arc of any radius centered
at A. Label the points of intersection P and
Q.
Step 2: Draw arcs of equal radius from P and Q. The line from A through their
intersection bisects BAC .
Comments:
Application: Making 45 and 22.5 angles from right angles, and making 30 and
15 angles from equilateral triangles.
Page 6 of 17
4. Other regular polygons
Application: Construct a regular pentagon.
Problem: Construct a regular pentagon.
Step 1
Step 2
Step 3
Step 4
Step 5
Step 6
Page 7 of 17
Step 7
Comments: s
Problem: Construct a regular pentagon using a framing square.
The distance from B to P is the length of each side of the pentagon.
Comments:
Page 8 of 17
Application: Construct a regular hexagon.
Problem: Construct a regular hexagon.
Just draw a circle, then mark the radius off six times around the circumference.
Comments: This forms the basis for many a church window. Extending the pattern
makes a pretty tiling.
Application: Construct a regular Octagon… just make a square inside a circle, then
bisect the angles. Or, construct 22.5 angles off of line segments until you get around
the octagon.
Page 9 of 17
5. Parallel Line construction
Problem: Construct a line parallel to a given line through a point not on the
given line.
Given line AB, and point C not on the line, draw line AC. Then, strike a random-length
arc centered at A that has intersection points D and E. Strike the same radius arc centered
at C that has intersection point F. Use a compass to measure the distance from D to E,
then mark off that same distance from point F to locate point G. The line through CG is
parallel to AB.
Comments:
Problem: Construct a line parallel to a given line through a point not on the
given line using the “draftsman's cheat”.
Comments:
Page 10 of 17
Problem: Partition a given line segment into a given number of congruent
segments.
To divide the original segment AB into 5 parts, draw a ray at any random angle, then
mark it off in 5 equal parts until you get to point C. Draw a line through BC, then
construct parallels to BC through each point on AC. The intersection points on AB will
divide it into 5 equal parts.
Comments: This is handy in carpentry for laying out stairs.
Application: Use this construction to lay out evenly spaced stairs on a stringer.
Page 11 of 17
Application: center line marking gauge
Page 12 of 17
Application: corner round-over
Page 13 of 17
Curves without strings
Application: Construct a circle without a string or trammel.
Problem: Draw a circle given its diameter using a framing square.
Use the fact that an angle inscribed across a diameter is 90.
Comments: This works better than a nail, string, and pencil.
Application: An ellipse using a special trammel, and an approximate arc using a
batten.
Page 14 of 17
6. Special proportions
Application: The Golden Ratio, as embodied in the Golden Rectangle, is the
most pleasing rectangle and proportion to the human eye. The ratio of the length to
1 5
:1  1.618 :1  8 : 5 . Use it when designing
the width of a Golden Rectangle is
2
furniture or buildings.
Problem: Construct a Golden Rectangle.
Start with square ABCD, then find the midpoint M of
side CD. Strike an arc centered at M that starts at A
and stops when it crosses ray CD (at point X).
Construct point Y perpendicularly above point X, and
the new rectangle BCXY is a golden rectangle.
Comments: This is very useful in designing furniture with nice proportions.
Page 15 of 17
Application: The Hambridge Progression is useful for designing chests of
drawers where the drawers get shorter and shorter near the top of the dresser.
Page 16 of 17
References
1. Tolpi, Jim, Measure Twice, Cut Once, Popular Woodworking Books, 1-55870428-0, 1996.
2. "Four Ways to Construct a Golden Rectangle," Fine Woodworking, Taunton
Press, February 2004, No. 168, p. 50.
3. Isaacs, I. Martin, Geometry for College Students, Brooks/Cole Thomson
Learning, 0-534-35179-4, 2001.
4. Posamentier, Alfred S. and William Wernick, Advanced Geometric
Constructions, Dale Seymour Publications, 0-86651-429-5, 1988.
5. Martin, George E., Geometric Constructions, Springer, 0-387-98276-0, 1998.
6. Sykes, Mabel, Source Book of Problems for Geometry, Dale Seymour
Publications, 0-86651-795-2, 1912.
7. Lundy, Miranda, Sacred Geometry, Wooden Books, 0-965-20578-9, 1998.
8. Bedford, John R., Graphic Engineering Geometry, Gulf Publishing Company, 087201-325-1, 1974.
9. Wingeom for Windows, website with freeware for geometric constructions (used
to create all diagrams in this packet) http://math.exeter.edu/rparris/wingeom.html
Page 17 of 17