Geometric Construction

In antiquity, geometric constructions of figures and lengths were restricted to the use of only a straightedge and compass (or in Plato's case, a compass only; a technique now called a Mascheroni construction). Although the term "ruler" is sometimes used instead of "straightedge," the Greek prescription prohibited markings that could be used to make measurements. Furthermore, the "compass" could not even be used to mark off distances by setting it and then "walking" it along, so the compass had to be considered to automatically collapse when not in the process of drawing a circle.

Because of the prominent place Greek geometric constructions held in Euclid's Elements, these constructions are sometimes also known as Euclidean constructions. Such constructions lay at the heart of the geometric problems of antiquity of circle squaring, cube duplication, and angle trisection. The Greeks were unable to solve these problems, but it was not until hundreds of years later that the problems were proved to be actually impossible under the limitations imposed. In 1796, Gauss proved that the number of sides of constructible polygons had to be of a certain form involving Fermat primes, corresponding to the so-called Trigonometry Angles.

Although constructions for the regular triangle, square, pentagon, and their derivatives had been given by Euclid, constructions based on the Fermat primes =17" border="0" width="30" height="14"> were unknown to the ancients. The first explicit construction of a heptadecagon (17-gon) was given by Erchinger in about 1800. Richelot and Schwendenwein found constructions for the 257-gon in 1832, and Hermes spent 10 years on the construction of the 65537-gon at Göttingen around 1900 (Coxeter 1969). Constructions for the equilateral triangle and square are trivial (top figures below). Elegant constructions for the pentagon and heptadecagon are due to Richmond (1893) (bottom figures below).

Given a point, a circle may be constructed of any desired radius, and a diameter drawn through the center. Call the center , and the right end of the diameter . The diameter perpendicular to the original diameter may be constructed by finding the perpendicular bisector. Call the upper endpoint of this perpendicular diameter . For the pentagon, find the midpoint of and call it . Draw , and bisect , calling the intersection point with . Draw parallel to , and the first two points of the pentagon are and . The construction for the heptadecagon is more complicated, but can be accomplished in 17 relatively simple steps. The construction problem has now been automated (Bishop 1978).

Simple algebraic operations such as , , (for a rational number), , , and can be performed using geometric constructions (Bold 1982, Courant and Robbins 1996). Other more complicated constructions, such as the solution of Apollonius' problem and the construction of inverse points can also accomplished.

One of the simplest geometric constructions is the construction of a bisector of a line segment, illustrated above.

The Greeks were very adept at constructing polygons, but it took the genius of Gauss to mathematically determine which constructions were possible and which were not. As a result, Gauss determined that a series of polygons (the smallest of which has 17 sides; the heptadecagon) had constructions unknown to the Greeks. Gauss showed that the constructible polygons (several of which are illustrated above) were closely related to numbers called the Fermat primes.

Wernick (1982) gave a list of 139 sets of three located points from which a triangle was to be constructed. Of Wernick's original list of 139 problems, 20 had not yet been solved as of 1996 (Meyers 1996).

It is possible to construct rational numbers and Euclidean numbers using a straightedge and compass construction. In general, the term for a number that can be constructed using a compass and straightedge is a constructible number. Some irrational numbers, but no transcendental numbers, can be constructed.

It turns out that all constructions possible with a compass and straightedge can be done with a compass alone, as long as a line is considered constructed when its two endpoints are located. The reverse is also true, since Jacob Steiner showed that all constructions possible with straightedge and compass can be done using only a straightedge, as long as a fixed circle and its center (or two intersecting circles without their centers, or three nonintersecting circles) have been drawn beforehand. Such a construction is known as a Steiner construction.

Geometrography is a quantitative measure of the simplicity of a geometric construction. It reduces geometric constructions to five types of operations, and seeks to reduce the total number of operations (called the "simplicity") needed to effect a geometric construction.

Dixon (1991, pp. 34-51) gives approximate constructions for some figures (the heptagon and nonagon) and lengths (pi) which cannot be rigorously constructed. Ramanujan (1913-1914) and Olds (1963) give geometric constructions for . Gardner (1966, pp. 92-93) gives a geometric construction for

Kochanski's approximate construction for yields Kochanski's approximation

Steinhaus (1999, p. 143). Constructions for are approximate (but inexact) forms of circle squaring.

Areas formula

(pi = = 3.141592...)

Area Formulas

Note: "ab" means "a" multiplied by "b". "a²" means "a squared", which is the same as "a" times "a".

Be careful!! Units count. Use the same units for all measurements. Examples

square = a ²

rectangle = ab

parallelogram = bh

trapezoid = h/2 (b₁ + b₂)

circle = pi r ²

ellipse = pi r₁ r₂

triangle =

one half times the base length times the height of the triangle

equilateral triangle =

triangle given SAS (two sides and the opposite angle)
= (1/2) a b sin C

triangle given a,b,c = [s(s-a)(s-b)(s-c)] when s = (a+b+c)/2 (Heron's formula)

regular polygon = (1/2) n sin(360°/n) S²
when n = # of sides and S = length from center to a corner

Units

Area is measured in "square" units. The area of a figure is the number of squares required to cover it completely, like tiles on a floor.

Area of a square = side times side. Since each side of a square is the same, it can simply be the length of one side squared.

If a square has one side of 4 inches, the area would be 4 inches times 4 inches, or 16 square inches. (Square inches can also be written in².)

Be sure to use the same units for all measurements. You cannot multiply feet times inches, it doesn't make a square measurement.

The area of a rectangle is the length on the side times the width. If the width is 4 inches and the length is 6 feet, what is the area?

NOT CORRECT .... 4 times 6 = 24

CORRECT.... 4 inches is the same as 1/3 feet. Area is 1/3 feet times 6 feet = 2 square feet. (or 2 sq. ft., or 2 ft²).

Prallelogram

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Parallelogram

A parallelogram is a quadrilateral with opposite sides parallel (and therefore opposite angles equal). A quadrilateral with equal sides is called a rhombus, and a parallelogram whose angles are all right angles is called a rectangle. And, since a square is a degenerate case of a rectangle, both squares and rectangles are special types of parallelograms.

The polygon diagonals of a parallelogram bisect each other (Casey 1888, p. 2).

The angles of a parallelogram satisfy the identities

			(1)
			(2)

and

(3)

A parallelogram of base and height has area

(4)

The height of a parallelogram is

(5)

and the polygon diagonals and are

			(6)
			(7)
			(8)
			(9)

(Beyer 1987).

The sides , , , and diagonals , of a parallelogram satisfy

(10)

(Casey 1888, p. 22).

The area of the parallelogram with sides formed by the vectors and is

			(11)
			(12)
			(13)

where is the two-dimensional cross product and is the determinant.

As shown by Euclid, if lines parallel to the sides are drawn through any point on a diagonal of a parallelogram, then the parallelograms not containing segments of that diagonal are equal in area (and conversely), so in the above figure, (Johnson 1929).

The centers of four squares erected either internally or externally on the sides of a parallelograms are the vertices of a square (Yaglom 1962, pp. 96-97; Coxeter and Greitzer 1967, p. 84).