Compass, Ruler, and Protractor Constructions - Sixth Grade Mathematics
Geometric constructions create exact spatial figures using only precision measuring tools, a compass for drawing arcs and circles, and a straightedge for drawing straight paths.
Essential Construction Instruments
Mathematical precision requires mastering the correct physical instruments:
1. Straightedge / Ruler: Used to draw straight lines, rays, and line segments, and to measure linear lengths in metric or customary units.
2. Compass: An instrument with two legs—one with a sharp pivot point and the other with a pencil point. It is used to draw circles, circular arcs of exact radius, and to transfer distances without measurement.
3. Protractor: A semicircular tool calibrated in degrees (0 to 180) used to measure and construct specific angles.
The Standard Compass:
Pivot Hinge
/\
/ \
Steel Needle / \ Pencil Lead
Point / \ Point
* *
Construction 1: Copying a Line Segment
To construct a segment congruent to segment AB without measuring numbers:
Step-by-Step Segment Duplication:
Original Segment AB:
A B
*======================*
Step 1: Draw a straight reference line with a straightedge and mark a starting point P.
Step 2: Place the steel compass needle on endpoint A and adjust the compass pencil so
the lead touches endpoint B exactly. The compass opening is now set to distance AB.
Step 3: Without changing the compass width, place the needle on starting point P.
Step 4: Draw a small arc crossing the reference line. Label the intersection point Q.
Result: Segment PQ is exactly congruent to Segment AB (PQ = AB).
Reference Line:
P Q (Arc mark)
<----*======================+-------->
Construction 2: Constructing a Perpendicular Bisector
A perpendicular bisector is a straight line that divides a line segment into two equal halves at an exact 90-degree right angle.
Constructing the Perpendicular Bisector of Segment XY:
C (Upper Arc Intersection)
*
/|\
/ | \
X / | \ Y
*------------+---+---+------------*
\ / | M | \ /
\ / | | \ /
\ / | | \ /
\ / | | \ /
*
D (Lower Arc Intersection)
Line CD is the Perpendicular Bisector of Segment XY:
1. Length XM = Length MY (M is the midpoint).
2. Line CD is perpendicular to Segment XY (Forms four 90° angles at M).
Procedure: Step 1: Place compass needle on point X. Open compass to more than half the length of XY. Step 2: Draw a large arc above and below segment XY. Step 3: Without changing compass width, place needle on point Y and draw intersecting arcs above and below. Step 4: Label the two points where the arcs cross as C and D. Step 5: Use a straightedge to draw a line connecting C and D. Line CD is the perpendicular bisector!
Construction 3: Bisecting an Angle
An angle bisector is a ray that divides an angle into two congruent, equal angles.
Bisecting Angle B:
A
/
/ D (Arc intersection)
/ .-*
/.' |
Ray BF -> /' |
/' |
Vertex B *'--------*--------- C
E
Ray BD bisects Angle B into two equal angles: Angle ABD = Angle DBC.
Procedure: Step 1: Place compass needle on vertex B. Draw an arc that crosses both sides of the angle at points D and E. Step 2: Place compass needle on point D and draw an arc in the interior of the angle. Step 3: Without changing compass width, place needle on point E and draw an arc intersecting the previous arc. Label the intersection point F. Step 4: Use a straightedge to draw Ray BF from vertex B through point F. Ray BF bisects the angle!
Construction 4: Drawing an Angle with a Protractor
To construct an angle of a specific measure, such as 65 degrees: Step 1: Draw a straight baseline ray with a straightedge, marking endpoint V. Step 2: Place the protractor center point over vertex V, aligning the baseline of the protractor with the ray. Step 3: Locate 65 degrees on the correct scale and make a small pencil tick mark on the paper. Step 4: Remove the protractor and use the straightedge to draw a ray from vertex V through the pencil mark.
Exercises
Exercise 1. Explain why the compass opening must be set to more than half the segment length when constructing a perpendicular bisector. What would happen if it were set to less than half?
Exercise 2. An angle measuring 76 degrees is bisected using a compass and straightedge. What is the degree measure of each of the two resulting angles?
Exercise 3. Describe the step-by-step procedure to construct a 90-degree angle using only a straightedge and compass, without a protractor.
Exercise 4. A student wants to find the exact midpoint of an 8.6 cm line segment using construction tools. Describe how the perpendicular bisector method achieves this without using ruler tick marks.
Solutions
Solution to Exercise 1. If the compass opening is less than half the length of the segment, the arcs drawn from opposite endpoints will never reach each other or intersect. The opening must be strictly greater than half the distance so that the two circular arcs overlap at two distinct points above and below the segment.
Solution to Exercise 2. Bisecting an angle divides it into two congruent halves. 76 / 2 = 38 degrees. Each of the two resulting angles measures 38 degrees.
Solution to Exercise 3. Step 1: Draw a straight line with a straightedge and mark a point P along it. Step 2: Use a compass centered at P to mark two equidistant points A and B on opposite sides of P, creating segment AB centered at P. Step 3: Open the compass wider than distance AP. Step 4: Place the needle on A and draw an arc above the line; place the needle on B with the same width and draw an intersecting arc at point C. Step 5: Draw a straight ray from P through C. Ray PC is perpendicular to the line, creating an exact 90-degree angle.
Solution to Exercise 4. By constructing the perpendicular bisector of the 8.6 cm segment, the line drawn through the arc intersection points crosses the original segment at its exact midpoint M. Because every point on the perpendicular bisector is equidistant from both endpoints, point M divides the segment into two equal segments of 4.3 cm each, locating the midpoint with pure geometric precision without relying on ruler millimeter marks.