D & M Pyramid Geometry

The Geomorphology and Geometry of the D&M Pyramid Erol O. Torun


The U.S. Viking mission to Mars in 1976 imaged some unusually shaped surface features in the Cydonia Mensae region. One of these features, a 5-sided pyramidal object, is examined with respect to its geological setting, its orientation, and its internal geometry. Study of geomorphological processes known to be active on Mars finds no plausible mechanism to explain the object's formation. The object exhibits bilateral symmetry, a complex internal geometry, and alignment with other anomalous landforms demonstrated to possess archaeoastronomical alignments. Taken together, the observations suggest that the object may have an artificial origin.


Among the images obtained by the Viking mission to Mars in 1976 are some that show peculiarly shaped surface features that are inconsistent with the regional geology and with surrounding landforms. Some of these objects are of such unusual morphology that there is considerable difficulty in theorizing a mechanism for their formation.

The first of these unusual features to be noticed was a 2 km long knob that resembled a human face staring straight up from the surface. The face was found in the Cydonia Mensae region in the northern hemisphere of Mars, and was officially dismissed by NASA as a trick of light and shadow. The face was rediscovered by DiPietro and Molenaar, computer scientists working at the Goddard Space Flight Center who unexpectedly found it while working with Viking imagery. They later found another image of the face that had been taken under different lighting conditions. Computer enhancement of these images revealed bilateral symmetry, detail resembling eyes, a nose, and a mouth, and persistence of this detail under two different sun angles. Their work was largely ignored by the planetary sciences community, and was published independently as a monograph (DiPietro and Molenaar [1]). Recent work by Carlotto [2] using single-image shape from shading techniques demonstrates that the face is not a trick of light and shadow or the result of variations in surface albedo. It is a three dimensional landform that, for whatever reason, has the form of a human face. Enhanced image processing by Carlotto more clearly reveals the presence of an eye socket in the shadowed side, as well as detail in the mouth that is suggestive of teeth.

DiPietro and Molenaar's image processing also aided in the observation of other landforms that are inconsistent with the local geology. Hoagland, seeing the work of DiPietro and Molenaar, began investigating the imagery and discovered the presence of a cluster of polyhedral objects, later named the "city", that have a rectilinear arrangement and a major axis aimed directly at the face (Figure 1). The face's axis of symmetry is itself perpendicular to the city's major axis. Hoagland [3] later demonstrated that a square arrangement of objects in the center of the city, termed the "city square", marks the exact midpoint along the city's major axis, and that a sightline from this vantage point to the eyes of the face once marked the summer solstice sunrise. This solstice alignment is satisfied once every million years, with the last alignment occurring ~1/2 million years ago.

In 1983, Hoagland organized and led the "Independent Mars Investigation", a cooperative effort of specialists in image processing, geology, architecture, and anthropology who studied these objects in greater detail. It was from this investigation that more infomation began to emerge concerning geometry and alignments. DiPietro and Molenaar had previously noted the presence of a massive pyramid, nearly 3 km in length and 1 km high, to the south of the city and face. Hoagland, working with a higher quality image processed by Stanford Research Institute, Inc., observed the object to be a 5-sided, bilaterally symmetrical pyramid whose axis of symmetry is aimed directly at the face. Hoagland also noted the alignment of one edge of the pyramid with the city square and of another edge of the pyramid with an unusually shaped round hill that lies to the east of the city on the same latitude as the city square. Hoagland named the object the "D&M Pyramid" to honor the earlier work of DiPietro and Molenaar.

The front of the D&M Pyramid (closest to the face) is formed by two congruent angles, with two larger congruent angles forming the sides. A fifth angle forms the rear section. The pyramid exhibits some domed uplift on its right side, and what appears to be an unusually deep impact crater further to the same side (Figure 2). The geometric regularity of the D&M Pyramid, together with its alignment with other enigmatic landforms, has led some to speculate that the object may have an artificial origin (DiPietro and Molenaar [1], Hoagland [3], Pozos [4]). Others discount this speculation, citing the slim likelihood of life evolving on Mars past the microbial stage, and the indeterminable likelihood of colonization by an extrasolar civilization.


The geology of Cydonia Mensae is described by Guest, Butterworth, and Greeley [5]. The region shows a mixture of smooth and fractured plains, and a small to moderate amount of cratering. Most relief in the vicinity of the D&M Pyramid is composed of mesas, knobs, and smooth plains material. Mesas are most likely the remnants of an earlier surface type that was removed by erosion, leaving mesas of more resistant material. Knobs may have been formed in a similar fashion, perhaps from rough, heavily cratered terrain. The shape of some knob material appears to have been modified by mass wasting or slumping, perhaps driven by the freezing and thawing of ground ice, with the excess material carried off by wind or, under different climatic conditions, by water or glacial ice.

Further evidence for some type of erosion is provided by the presence of several pedestal craters in Cydonia Mensae. A pedestal crater is an impact crater surrounded by an ejecta blanket that ends in a steep scarp that may drop hundreds of meters to the surface. The ejecta blanket is presumably composed of material that is more resistant to erosion than the surrounding surface.

The morphology of Cydonia Mensae is thus complex and not completely understood. The region exhibits evidence for previous epochs of cratering, erosion, and deposition, contributing to the wide variety of observed landforms and surface types.


Viking orbiter frames 35A72, 70A11, and 70A13 show the D&M Pyramid, located at 40.65N 9.55W. All three frames were taken close to periapsis and yield a pixel resolution of ~50 m. Other frames show the region at resolutions insufficient for detailed study. The frame selected for the examination of the D&M Pyramid's geometry is 70A13. (The nomenclature indicates the 70th orbit, 13th frame of the "A" orbiter.) This frame was taken at a higher sun angle than 35A72 and thus shows more of the object's structure. Images of 70A13 used for angle measurements were an enhanced closeup of the D&M Pyramid prepared by Carlotto, (Figure 2) and an NGF filtered orthographic rectification of the entire frame obtained from the National Space Science Data Center (USA).


The hypothesis that the D&M Pyramid is the result of geomorphological processes requires that some mechanism be proposed to account for its formation. The processes discussed below are evaluated in terms of their ability to produce landforms with the same general morphology as the object under study. Processes meeting this requirement are then discussed with respect to the specific morphology of the D&M Pyramid.

4.1 Fluvial Deposition/Erosion

It is now almost universally accepted that liquid water once flowed on Mars, this belief supported by the discovery by the Mariner 9 mission [6] of numerous drainage channels whose form is consistent with formation by water (as opposed to lava channels, which have also been found in some areas). Fluvial processes can yield roughly symmetrical relief, such as the teardrop-shaped islands observed in many Martian stream beds. But in this case, fluvial processes can be ruled out as mechanisms for forming the D&M Pyramid as there are no indications that water ever flowed 1 km deep in Cydonia Mensae. It is also true that sharp edged multi-faceted symmetrical shapes are not characteristic of fluvial landforms.

4.2 Aeolian Deposition/Erosion

Aeolian depositional landforms on Mars typically take the form of sand dunes which are morphologically similar to those found on Earth, and have been described by Breed, Grolier, and McCauley [7]. Dunes may occur singly, in small groups, or in dune fields, and are transient in form and location. Dunes will occasionally form star-shaped dunes and crescent-shaped symmetrical dunes known as barchans, but no dune will ever form a symmetrical polyhedron resembling the one under study. Flat sides and straight edges are unobserved in terrestrial or Martian sand dunes.

Aeolian erosion is now the dominant mechanism of geomorphic change on Mars and has been since the disappearance of liquid water and the cessation of volcanic activity, both of which have been estimated to have occurred early in the planet's history

(>2 b.y. ago). One type of aeolian erosional feature found on Mars that can assume roughly pyramidal shapes are known as yardangs (Ward [8]). Yardangs are landforms that have been modified by deflation (removal of particles by wind) and by the abrasive action of wind-borne particulates, and typically resemble hills with sharp edges aligned with the prevailing winds. Their shape has been likened to upturned boat hulls. Yardangs can occasionally exhibit sharp edges, roughly flat sides, and bilateral symmetry.

Comparison of the D&M Pyramid with landforms known to be yardangs immediately reveals some serious inconsistencies. The D&M Pyramid is an isolated landform with no other nearby objects exhibiting a similar shape and orientation. Yardangs normally begin with the downcutting and expansion of low spots, producing a series of parallel ridges that are gradually denuded into separate hills. These hills are eroded further by abrasion on the windward side and by deflation on the leeward side. It is rare for yardangs to be found in isolation.

Another type of aeolian landform that can be somewhat pyramidal in shape are known as ventifacts. Terrestrial ventifacts are normally formed from small rocks that are exposed to the abrasive action of sand carried by the wind. Multifaceted terrestrial ventifacts are believed to have been produced as a result of movement of a rock causing it to present different faces to the direction of the prevailing winds, a process described by Sharp [9] and Greeley and Iverson [10]. Large ventifacts can also exist, produced from boulders and assuming a roughly pyramidal shape with three edges (dreikanters). These landforms present a long edge toward the prevailing winds, and a somewhat flat surface in the opposite direction. The leading edge is cut by abrasion of wind-borne particulates and the trailing surface apparently formed by deflation from locally reversed airflow, mechanisms essentially the same as those that form yardangs.

Five-sided symmetrical ventifacts or yardangs appear to be totally nonexistent on Earth and Mars. Prevailing winds are not likely to have shifted periodically with perfect symmetry and timing. Even if this seemingly impossible condition were satisfied, another factor would prevent such an object from forming. As noted above, locally reversed airflow can cut a flat surface perpendicular to the wind direction on the leeward side of a wind-cut hill. This locally reversed airflow, and associated suface level turbulence, would prevent the formation of this hypothetical five-sided ventifact. Each time the wind shifted to a new direction, the reversed airflow would start erasing the edges formed by other wind directions. The end result would not be a pyramidal hill, but rather a round one.

The overall morphology of the D&M Pyramid, with its straight edges and flat surfaces in radial arrangement, is inconsistent with the morphology of aeolian landforms. The nearby face shows no evidence of wind faceting, and there are no intervening objects between the face and the D&M Pyramid to deflect wind. Also inconsistent is the presence of a flat faced protuberance at the front of the object, a flat surface that should not exist at the leading edge of wind cut features such as yardangs or ventifacts. It is reasonable to conclude that aeolian processes cannot have produced the D&M Pyramid due to the lack of a plausible mechanism of formatge amounts of rainfall in recent epochs. Additionally, the D&M Pyramid has no vent at its apex, and exhibits a symmetry unknown in volcanic landforms.

4.5 Crystal Growth

Nearly all examples of naturally occurring symmetrical polyhedra are the result of crystal growth. It has been theorized that a freeze-thaw cycle might have produced unusual crystal growth from the water believed by some to lie beneath the surface of Mars, but the D&M Pyramid could not possibly have been produced in this manner. The size of the object (> 2 km long) would alone preclude the possibility of it being an ice crystal, as would the fact that ice crystals have hexagonal symmetry. Crystallization of other substances can also be ruled out as it is impossible for any crystal to possess 5-fold symmetry (Bunn [12]). The D&M Pyramid actually deviates from perfect 5-fold symmetry by being somewhat elongated along one axis, and is thus even further removed from crystalline geometries.

The Geomorphic Hypothesis is thus left with no mechanism that can explain the formation of the D&M Pyramid. This object's 5-sided shape and bilateral symmetry is unlike any landform seen to date in this solar system, and even small-scale phenomena such as crystal growth cannot explain its morphology.

All observations to date of the geophysics of Mars, its gravity, meteorology, geomorphology, ect., indicate that Mars is a place where the laws of physics and principles of geomorphology as we understand them apply, with minor variations due to gravity and atmospheric density and content. It is illogical to assume that there is one small place on the surface of Mars where these same principles are being violated. Being thus faced with no known natural mechanism to account for the D&M Pyramid's formation, other possible mechanisms need to be explored.

While the above discussion of geomorphological processes may appear tedious, especially the wild notion that crystal growth might be responsible for this object, this degree of examination is necessary. There is a natural tendency to jump to hasty conclusions regarding objects that appear to have an extraordinary origin. After it has been shown that no known natural mechanism can be responsible, only then, as a last resort, can we justifiably examine the possibility of intelligent design.


The hypothesis that the D&M Pyramid may be the product of intelligent design cannot be blindly accepted simply because there is no geomorphological explanation, but must be subjected to an objective analysis, especially due to its location on Mars, a place where life is not known to have existed. Thus the Intelligence Hypothesis cannot be explored without first answering a preliminary question: What are the hallmarks of architectures that distinguish them from landforms and how may they be objectively recognized and evaluated?

In searching for the signs of intelligent design, we cannot allow ourselves any subjective, Earth-based frames of reference. The observation "it looks like a pyramid" is sufficient to focus our attention towards further investigation, but proves nothing else. With subjective appearances set aside, there are a number of objective characteristics of architectures that can be identified:

  1. Is the object's geometry inconsistent with known landforms and geomorphological processes? (i.e. does the object exhibit straight lines, curved lines having fixed radii, regular patterns, one or more axes of symmetry, and does the combination of these characteristics preclude geomorphology as a mechanism of origin?)
  2. Is the object aligned with the cardinal directions and/or with significant astronomical events?
  3. Is the object collocated with other objects that are also inconsistent with the surrounding geology? And if so, are they geometrically aligned with each other?
  4. Does the object's geometry express mathematically significant numbers, and if so, do the numbers fit any identifiable pattern?

The characteristics listed above cannot by themselves establish an object as being the product of intelligent design. What is required is a totality of geometric relationships that, when viewed as a whole, preclude the likelihood of a natural origin. A simple example of this sort of epistemological problem can be illustrated by imagining that one is exploring a terrestrial limestone cave. While it is possible that one may eventually encounter a stalagmite that vaguely resembles a human face, one will never encounter the Washington Monument in scale miniature. That degree of mathematical precision does not occur in nature in that environment and from those materials, and would be properly interpreted as being the product of intelligent design.

5.1 The Geometry of the D&M Pyramid

5.1.1 Methods

In exploring the geometry of the D&M Pyramid, the most conservative approach possible was pursued. The reasons for this are partly historical. The Great Pyramid of Egypt has a long history of being mathematically abused by mostly well-meaning investigators who, in attempts to "prove" various theories, have subjected the Great Pyramid to excessively complex measurements and calculations. If the number of such measurements combined at any one time is large enough, it is possible to "discover" almost any number desired.

Study of the D&M Pyramid's geometry was therefore restricted to three types of relationships:

  1. Examining the values of observable angles expressed in radian measure.
  2. Testing the ratio of one of the observable angles with another such angle for equality with mathematically significant numbers.
  3. Examining the Sine, Cosine, and Tangent of measured angles for the presence of mathematically significant numbers.

These approaches were selected due to their simplicity, their validity in number bases other than decimal, and their independence from our convention of expressing angles as a portion of a 360 degree circle.

5.1.2 Analysis

As previously mentioned, the D&M Pyramid shows signs of being damaged on one side, perhaps by a meteoric impact. Some angle reconstruction was therefore necessary before any accurate measurements could be obtained. The following technique was applied to the Carlotto image in preparing an overlay of all visible angles:

  1. The most distinct edges on the pyramid, those on the sunlit side, were extended to locate the position of the original apex.
  2. A straight line was drawn from the apex through the flat protuberance at the front of the pyramid to mark the axis of symmetry.
  3. A line was extended from the apex to the right front corner, which is clearly visible on the Carlotto image.
  4. The figure was enclosed, based upon the left side of the pyramid and the right front corner.

An NGF filtered orthographic negative of Viking orbiter frame 70A13 was obtained from the National Space Sciences Data Center. Because the data collected were primarily angle measurements, the orthographic projection was necessary to ensure that the geometry of the object under study was accurately represented on the image. While the quality of this batch processed image is far lower than the Carlotto image, it was sufficient for direct projection. This orthographic image was measured to confirm the observation that the Carlotto image is so nearly orthographic that any errors due to the imaging geometry are negligible.

All visible angles of the D&M Pyramid were measured (? 0.2?) and subjected to the tests mentioned earlier: radian measure, angle ratios, and trig functions. The results of these tests are listed on Figure 3, together with a labeled diagram of the pyramid's angles as viewed from above. The accuracy of the tests are evaluated in Table 1. Figure 4 shows frame 35A72 in its entirety, displaying the relative location of the D&M Pyramid, the city, and the face. Geographic positions are indicated by the coordinate grid, produced by Davies from data collected for a control network for Mars [13].

5.1.3 Evaluation

As can be seen from Figure 3, the geometry of this object is logical and mathematically elegant. Some of the most significant observations are:

  1. The reconstructed geometry of the D&M Pyramid exhibits perfect bilateral symmetry, with a pair of congruent angles forming the front, and another pair of congruent angles forming the sides. The rear, fifth section is bisected by the pyramid's axis of symmetry thereby forming a sixth, "virtual" section labeled E on the diagram.
  2. The radian measure of many angles are equal to mathematically significant numbers.
  3. The trig functions that were examined also generated mathematically significant numbers for many of the angles.
  4. The ratio of any two dissimilar apex angles is equal to a mathematically significant number. This is also true for many angle ratios involving the ground level angles, labeled F, G, and H.

The geometric relationships found in the D&M Pyramid are tabulated in Figure 3 according to their mode of expression: radian measure, angle ratios, and trig functions. The analysis shows that the geometry of the D&M Pyramid expresses the square roots of 2, 3, 5, and the constants e and pi. The existence of these numbers would have been known by any civilization capable of designing the Cydonia complex. The mathematical importance of these numbers, and the modes by which they are expressed, are independent of our convention of counting in the decimal system, as well as our convention of subdividing circles into 360 degrees.

Of greater significance than the mere presence of these numbers is the fact that most of them were expressed by all three methods used to analyze the D&M Pyramid. The numbers e, pi, the square root of 5, and various expressions of 3 appear with radian measure, angle ratios, and trig functions. This is important when one realizes that the hypothetical architects of an object such as this could not have known what techniques might be applied to analyze its geometry. If the geometry of the D&M Pyramid was intended to display certain numbers, the expression of these numbers via numerous methods would have been seen as increasing the likelihood of their discovery.

The question of whether these numbers are the ones intended for expression needs to be addressed. It is true, for example, that (sqrt 3)/2 closely approximates the number e/pi, and there are no doubt other numbers that approximate the other relationships. The case for e, pi, and sqrt 5 being the intended numbers is eloquently made by the fact that all possible combinations of these numbers are represented in the geometry of the D&M Pyramid: e-pi, e-sqrt 5, and ?-sqrt 5. These combinations are shown, grouped by value, on the table in Figure 5. The identity of these numbers is further underscored by the fact that the D&M Pyramid is itself 5-sided, and the extreme redundancy of the number 3.

An even more provocative observation is that there appear to be reasons for these numbers and mathematical constants appearing where they have been found. These observations are important to understanding the geometry of the D&M Pyramid in context with the other enigmatic objects in Cydonia Mensae; if the objects are artificial, such logic may provide insight into the intentions of the architects. Consider the following relationships:

  1. Angle C and angle A share something in common; there are two sections on the D&M Pyramid that form each angle. The quotient of angle C divided by angle A equals sqrt 2.
  2. Angle B, which subtends one third of a circle, forms a perfectly symmetrical "trident" aimed at the face. This angle, divided by angle D which is exactly opposite angle B along the pyramid's axis of symmetry, equals sqrt 3. The tangent of angle B also equals the negative of sqrt 3. Hoagland has recently observed that the pervasive "redundancy of 3's" may be symbolic of the earth, the third planet from the sun and the place where the life form resembling the face in Cydonia can be found.
  3. The relationships between e and sqrt 5 may also be suggestive of biology. As noted in the discussion of geomorphology, 5-sided symmetry is not characteristic of nonliving systems. Life forms on earth, however, often exhibit 5sided symmetry, especially in the Plant Kingdom. The constant e, the base of the natural logarithm, is also known as the law of organic growth. It is a way of describing growth where the increment of growth is always proportional to the size of the growing quantity, as is often the case in biological systems. Most formulae devised for the study of organic growth, whether for population studies, or predictions of microbial and plant growth, incorporate the number e as a factor. The relationship between e and sqrt 5 might therefore be interpreted as being symbolic of "the exponential growth of life".

The interpretation of these numbers as a biological metaphor is supported by the fact that the D&M Pyramid possesses another characteristic of living things - bilateral symmetry, and by the alignment of the D&M Pyramid's axis of bilateral symmetry with the one object in Cydonia Mensae that most closely resembles a living thing: the face.

6. Conclusions

There are only two possible categories into which theories for the origin of the enigmatic objects at Cydonia Mensae fit: either they are natural or they are artificial. It is the author's opinion that the case for an artificial origin for these objects cannot rest alone upon alleged similarities with terrestrial architectures or "humanoid" faces. The human mind has great capabilities for pattern recognition and an even greater imagination. Instead, the case for artificiality must rest on the geometry of what can be observed. In this way, we can measure the objects we see and note the existence of symmetry, geometric relationships, and mathematical constants. The more complex and logical these relationships are, and the more frequently mathematical constants are expressed, the greater is the likelihood that they are there by design.

This investigation of the D&M Pyramid reveals a morphology that is inconsistent with the surrounding geology. The geomorphological processes observed to exist on Mars not only fail to provide a potential mechanism for the D&M Pyramid's formation, but seem to preclude its very existence. Analysis of the object's geometry, and its alignment with other anomalous landforms, reveal intricate relationships that are numerous and logical, and are suggestive of highly sophisticated design. While a mechanism of origin for the D&M Pyramid cannot be stated with absolute certainty at this time, one fact is very clear: according to our current knowledge, something in Cydonia Mensae is extremely wrong. Either we are looking at the ruins of a civilization, or our knowledge of geomorphology as it relates to the geophysics of Mars is in need of revision. These questions should be answered by the Mars Observer mission, to be launched in 1992. Mars Observer will carry a camera capable of better than 2 m ground resolution, as compared to 50 m for Viking. Whatever the answers returned by this mission are, they are guaranteed to be extraordinary.

The author wishes to acknowledge Dr. Mark J. Carlotto, The Analytic Sciences Corporation; Dr. Merton E. Davies, RAND Corporation; The Mars Project; and the National Space Science Data Center for the imagery and coordinate information, and Mr. Richard C. Hoagland for background information and collaboration on current research. The views and opinions expressed in this paper are those of the author and not necessarily those of the Defense Mapping Agency Systems Center or the Department of Defense.


1, V. DiPietro and G. Molenaar, 'Unusual Martian Surface

Features' Mars Research, Glen Dale, Maryland (1982).

2. M.J. Carlotto, 'Digital imagery analysis of unusual Martian

surface features', Applied Optics, 27, 1926-1933 (1988).

3. R.C. Hoagland, 'The Monuments of Mars - A City on the Edge of

Forever', North Atlantic Books, Berkeley, 1987.

4. R.R. Pozos, 'The Face on Mars: Evidence for a Lost

Civilization?', Chicago Review Press, Chicago, 1986.

5. J.E. Guest, P.S. Butterworth and R. Greeley, 'Geological

Observations in the Cydonia region of Mars from Viking',

J. Geophys. Res., 82, 4111-4120 (1977).

6. National Aeronautics and Space Administration, 'Mars as

Viewed by Mariner 9' (NASA SP-329), Scientific and Technical

Information Office, Washington, D.C. (1976).

7. C.S. Breed, M.J. Grolier and J.F. McCauley, 'Morphology and

distribution of common 'sand' dunes on Mars: comparison with

the Earth', J. Geophys. Res., 84, 8183-8204 (1979).

8. A.W. Ward, 'Yardangs on Mars: evidence of recent wind

erosion', J. Geophys. Res., 84, 8147-8166 (1979).

9. R.P. Sharp, 'Pleistocene ventifacts east of the Big Horn

Mountains, Wyoming', J. Geol., 175-195 (1948).

10. R. Greeley, and J.D. Iverson, 'Wind as a Geological Process

on Earth, Mars, Venus, and Titan', Cambridge University

Press, Cambridge (1985).

11. M.H. Carr, 'The Surface of Mars', Yale University Press, New

Haven (1981).

12. C. Bunn, 'Crystals: Their Role in Nature and in Science', pp.

84-87, Academic Press, New York and London (1964).

13. M.E. Davies and F.Y. Katayama, 'The 1982 Control Network of

Mars', J. Geophys. Res., 88, 7503 (1983).

(The diagram that follows is an extended ASCII version of

Figure 3. Some of the other figures cited in the text can also be found in the book 'Monuments of Mars' (Hoagland [3]). Figures 1, 2, and 4 correspond to Plates 7, 14, and 9, respectively.)

The D&M Pyramid


* *


* *


* *


F * B *


* A *


* C + *

* *


* * E * *


* D *

G |

* *

H |

* * * * * * *


? Angles ? Angle Ratios ? Trig. Functions ? ???????????????????????????????????????????????????????????

? degrees ? radians ? C/A = ?2 ? TAN A = ?3 ?

????????????????????? B/D = ?3 ? TAN B = -?3 ?

? A=60.0 = ?/3 ? C/F = ?3 ? SIN A = e/? ?

? B=120.0 = 2?/3 ? A/D = e/? ? SIN B = e/? ?

? C=85.3 ? C/D = e/?5 ? TAN F = ?/e ?

? D=69.4 = e/?5 ? A/F = e/?5 ? COS E = ?5/e ?

? E=34.7 ? H/G = e/?5 ? SIN G = ?5/? ?

? F=49.6 = e/? ? B/C = ?/?5 ? ?

? G=45.1 ? D/F = ?/?5 ? ?

? H=55.3 ? ? ? ?????????????????????????????????????????????????????????????

Figure 3

Table 1

RELATIONSHIP IDEAL OBSERVED % ERROR ------------------------------------------------------

Angle Ratios:

C/A ?2 = 1.41 1.42 0.71

B/D ?3 = 1.73 1.73 0.00

C/F ?3 = 1.73 1.72 0.58

A/D e/? = 0.865 0.865 0.00

C/D e/?5 = 1.22 1.23 0.82

A/F e/?5 = 1.22 1.21 0.82

H/G e/?5 = 1.22 1.23 0.82

B/C ?/?5 = 1.40 1.41 0.71

D/F ?/?5 = 1.40 1.40 0.00

Trig. Functions:

TAN A ?3 = 1.73 1.73 0.00

TAN B -?3 = -1.73 -1.73 0.00

SIN A e/? = 0.865 0.866 0.12

SIN B e/? = 0.865 0.866 0.12

TAN F ?/e = 1.16 1.17 0.86

COS E ?5/e = 0.823 0.822 0.12

SIN G ?5/? = 0.712 0.708 0.56

Radian Measure:

A ?/3 = 1.05 1.05 0.00

B 2?/3 = 2.09 2.09 0.00

D e/?5 = 1.22 1.21 0.82

F e/? = 0.865 0.866 0.12

Mean Error = 0.36% ; Standard Deviation of Error ?x = 0.36 %