Classical Constructions:

In relation to Geometry, the term "construction" refers to the drawing of geometric figures using only a compass and a straightedge (an unmarked ruler). An amazingly large number of constructions can be made using only this minimal set of tools.
Note: There are three famous constructions that are not possible using only a straightedge and compass:
• trisecting an angle (given an angle, construct an angle whose measure is exactly one-third of the given angle)
• squaring a circle (construct a square that has exactly the same area as a circle)
• doubling a cube (given the side of a cube, construct a side length for a cube that will have exactly double the area of the given cube)

The beauty of a construction is that the relationship it portrays can be supported by a formal proof. Once a construction is drawn, a geometric proof can be supplied to show that the construction actually behaves as expected. You can think of constructions as physical verifications of geometric concepts.

Constructions can be produced using a variety of tools and methods, such as compass and straightedge, string, reflective devices, paper folding, and dynamic geometric software. The most common of these choices is the compass and straightedge, which will be the construction method used at this web site. Other construction methods will be demonstrated by example only.

Taking measurements using a protractor or a ruler is never considered to be part of a geometric construction. Also, freehand drawing is never part of a geometric construction, except for labeling of the finished drawing.


The Compass:


A compass is used for drawing circles and arcs, and for measuring distances. While this tool comes in a variety of styles, most compasses have two legs, one with a spike (point) and the other with a pencil, pen, lead, chalk, or other drawing utensil. The legs of a modern day compass are hinged to allow for opening and closing, and the compass may be tightened to maintain a specific length.
Note: This tool is used for drawing and is not the navigational device used for determining north, south, east and west directions.


Historical note: In 300 BC, constructions were made using an Euclidean compass which was not capable of being used to copy a length, since it collapsed after each use. Any construction done with an Euclidean compass can be done with our modern compasses which can be set to hold a specific length.

It is interesting to note that computer-aided programs have nearly replaced the use of hand-held compasses. Today, the hand-held compass is primarily used in the teaching of geometrical concepts and technical drawing.

Hints for using a compass:
• Be sure to use a sharpened pencil.
If your compass has a lead insert, use sandpaper to sharpen the lead.

• If your compass requires that you insert a pencil, use a small pencil (short) as it will make it easier to balance the compass.

• Be sure to adjust your compass so that the leg with the sharp point and the leg with the pencil lead are the same length. Close the compass to see if the two legs are the same length.

• Place several sheets of paper under your worksheet. Allowing the compass point to pierce the papers will help stabilize the compass and prevent it from slipping.

• Hold the compass lightly and try to keep your wrist flexible.

• If you have trouble moving your wrist when drawing circles, try rotating the paper under the compass.

• Try to maintain a constant, but light, pressure on the compass. Do not press down too hard on the paper.



The Straightedge:



A straightedge is a tool used for drawing straight lines, rays or segments.

While a straightedge is typically a clear plastic tool in the shape of a triangle and devoid of markings, it technically can be any item which will guide your pencil when drawing a straight line. A ruler may be used as a straightedge as long as the markings on the ruler are ignored.




Required Constructions:



The basic constructions stated "by name" in the Common Core Standards and PARCC are listed below. While it is essential to know how to develop these specific constructions, it is also important to know how to use these constructions to produce other diagrams illustrating (or supporting) geometric definitions and concepts. In other words, you need to be able to apply these basic skills to new situations and new constructions.




Construct an equilateral triangle.
Construct a square.
Construct a regular hexagon inscribed in a circle.


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