Geometry, Tools of

Plane (or Euclidean) geometry is the branch of mathematics that studies figures (such as points, lines, and angles) constructed only with the use of the straightedge and the compass. It is primarily concerned with such problems as determining the areas and diameters of two-dimensional figures. To determine geometric designs four important tools of geometry—compass, straightedge, protractor, and ruler—are used. Technically a true geometric construction with Euclidian tools, originally used by the ancient Greeks, uses only a compass or a straightedge. The ruler and protractor were later inventions. Today, the study of geometry is an essential part of the training of such professionals as mathematicians, engineers, physicists, architects, and draftspersons.

As early as 2000 b.c.e. geometers were concerned with such problems as measuring the sizes of fields and irrigation systems, and laying out accurate right angles for corners of buildings and monuments. Greek mathematician Euclid (c. 300 b.c.e.) and other geometers formalized the process of building geometrical figures with the use of specific tools. The ancient Greeks introduced construction problems that required a certain line or figure to be constructed by the use of the straightedge and compass alone. Simple examples are the construction of a line that will be twice as long as another line, or of a line that will divide a given angle into two equal angles.

Basic Tools

Straightedge. A straightedge is a geometric tool used to construct straight lines. It contains no marks. As the name says, it is a "straight edge." A ruler can be used as a straightedge by simply ignoring the measuring marks on it.

Ruler. A ruler is a geometric tool used to measure the length of a line segment. A ruler is basically a straightedge with marks usually used for measuring either inches or centimeters. To use a ruler, place the zero mark on the point to begin the measurement. To stop measuring, look at the mark on the ruler that lies over the point at which the measurement is to end.

Compass. A compass is a V-shaped tool used to construct circles or arcs of circles. (See sketch below on left.) One side of the "V" holds a pencil and the other side is a point. The point anchors the compass at one location on the paper as the tool is turned so the pencil can trace circles, arcs, and angles. A compass is adjustable: the setting determines how far away from the point the arc would be located. Once the user determines the correct setting the compass is turned around its anchoring point so that the pencil creates a mark, an arc, or a full circle.

Protractor. A protractor is a geometric tool in the shape of a semicircular disk, as shown above on the right. It is used to measure the size of an angle in degrees—usually from 0 to 180 degrees. To use a protractor, lay the protractor on the angle to be measured. There will be a mark on the bottom of the protractor (the straight edge of it) indicating its middle. Place this mark over the origin of the angle, and align the straight edge of the protractor with one side of the angle. Where the other side of the angle intersects the protractor there will be a mark with a number next to it. This measurement is the measure of the angle.

Solving Construction Problems

Construction problems are generally solved by following six steps.

1. Provide a general statement of the problem that describes what is to be constructed.
2. Draw a figure representing the given parts.
3. Provide a statement of what is given in step 2.
4. Provide a specific statement of the result to be obtained.
5. Develop the construction, with a description (reason) for each step.
6. Provide a statement proving that the desired result was obtained.

As an example using a straightedge: Given line segments a and b, construct the line segment c = a + 2b.

This example is solved as follows. On working line w, draw a line segment equal to the length of line segment a. At the right end of a draw a line segment equal to length of line segment b. At the right end of line segment a + b, draw a second line segment equal to length of b. The resulting line segment is the desired line segment c = a + b + b = a + 2b.

As an example of using a compass: Construct an angle equal to the given figure ∠BAC with line segments AB and AC.

This example is solved as follows. In Step One, place the compass pivot point at A and adjust the compass width to be between A and B, then construct an arc that intercepts both line segments AB and AC. Leaving the compass width unchanged, draw a line w and place the pivot point at an arbitrary point A. Pivot the compass from line w , forming an arc like in Step One (as shown in Step Two). The arc's intersection with line w is denoted B′.

Referring to Step One, now adjust the compass width to equal the distance between the arc's intersection with AB and AC. With this width, move the compass down to Step Two, place the pivot point at B ; and make an arc that intersects with the first arc, as in Step Three. The line segments A'C' and and A'B', as shown in Step Four, complete the new angle ∠B′A′C′ that is equal to the original angle ∠BAC.

William Arthur Atkins with

Philip Edward Koth

Bibliography

Boyer, Carl B. A History of Mathematics, 2nd ed. New York: John Wiley & Sons, 1991.

Henderson, Kenneth B., Robert E. Pingry, and George A. Robinson. Modern Geometry: Its Structure and Function. New York: Webster Division, McGraw-Hill, 1962.

Ulrich, James F., Fred F. Czarnec, and Dorothy L. Guilbault. Geometry. New York: Harcourt Brace, 1978.