Pyramid

Resources

A pyramid is an artificially made geometric solid of the shape made famous by the royal tombs of ancient Egypt. It is a solid with a base of a three-sided or four-sided polygon (trilateral or quadrilateral, respectively) and whose lateral faces are triangles (trilateral sides) with a common vertex (the vertex of the pyramid). The measurements of these triangles classify the shape as usually isosceles but sometimes as equilateral. In the case of the Egyptian pyramid of Cheops, the base is an almost perfect square 755 ft (230 m) on an edge, and the faces of triangles that are approximately equilateral.

Some Egyptian ancient pyramids are the Pyramid of Khufu (Great Pyramid) at Giza, the Pyramid of Neferirkare Kakai at Abu Sir, and the Step Pyramid of Djozer at Saqqara. Other ancient pyramids are located in China, France, South America, Mexico, the Ukraine, Sudan, and Bosnia-Herzegovina.

The base of a pyramid can be any polygon of three or more edges, and pyramids are named according to the number of edges in the base. When the base is a triangle, the pyramid is a triangular pyramid. It is also known as a tetrahedron since, including the base, it has four faces. When these faces are equilateral triangles, it is a square pyramid, having a square as its base.

The pyramids most commonly encountered are regular pyramids. These pyramids have a regular polygon for a base and isosceles triangles for lateral faces. Not all pyramids are regular, however.

The height of a pyramid can be measured in two ways, from the vertex along a line perpendicular to the base and from the vertex along a line perpendicular to one of the edges of the base. This latter measure is called the slant height. Unless the lateral faces are congruent triangles, however, the slant height can vary from face to face and will have little meaning for the pyramid as a whole. Unless the word slant is included, the term height (or altitude) refers to the height.

If in addition to being congruent, the lateral faces are isosceles, the pyramid will be regular. In a regular pyramid, right triangles are to be found in abundance. Suppose one has a regular pyramid whose altitude is VC and slant height VD. Here the triangles VCD, VDE, VCE, and CDE are all right triangles. If in any of these triangles one knows two of the sides, one can use the Pythagorean theorem (whose equation form is: a² + a² = c², where c is the length of the hypotenuse and a and b are the lengths of the other two sides) to figure out the third. This, in turn, can be used in other triangles to figure out still other unknown sides. For example, if a regular square pyramid has a slant height of two units and a base of two units on an edge, the lateral edges have to be 5 units and the altitude 3 units.

There are formulas for computing the lateral area and the total area of certain special pyramids, but in most instances it is easier to compute the areas of the various faces and add them up.

Volume is another matter. Figuring volume without a formula can be very difficult. Fortunately, there is a rather remarkable formula dating back at least 2,300 years.

In Proposition 7 of Book XII of his Elements, Greek mathematician Euclid of Alexandra (c. 325–c. 265 BC showed that “Any prism which has a triangular base is divided into three pyramids equal to one another which have triangular bases.” This means that each of the three pyramids into which the prism has been divided has one-third the prism’s volume. Since the volume of the prism is the area, B, of its base times its altitude, h: the volume of the pyramid is one third that, or Bh/3.

Pyramids whose bases are polygons of more than three sides can be divided into triangular pyramids and Euclid’s formula applied to each. Then if B is the sum of the areas of the triangles into which the polygon has

KEY TERMS

Altitude— The distance from the vertex, perpendicular to the base.

Pyramid— A solid with a polygonal base and triangular lateral faces.

Slant height— The distance form the vertex, perpendicular to the edge of the base.

been divided, the total volume of the pyramid will again be Bh/3.

If one slices the top off a pyramid, one truncates it. If the slice is parallel to the base, the truncated pyramid is called a frustum. The volume of a frustum is given by the curious formula (B + B′ + √BB′)h/3, where B and B′ are the areas of the upper and lower bases, and h is the perpendicular distance between them.

Resources

BOOKS

Burton, David M. The History of Mathematics: An Introduction. New York: McGraw-Hill, 2007.

Schoch, Robert M. Voyages of the Pyramid Builders: The True Origins of the Pyramids, from Lost Egypt to Ancient America. New York: Meremy P. Tarcher/Putnam, 2003.

Setek, William M. Fundamentals of Mathematics. Upper Saddle River, NJ: Pearson Prentice Hall, 2005.

J. Paul Moulton

Pyramid

A pyramid is a geometric solid of the shape made famous by the royal tombs of ancient Egypt. It is a solid whose base is a polygon and whose lateral faces are triangles with a common vertex (the vertex of the pyramid). In the case of the Egyptian pyramid of Cheops, the base is an almost perfect square 755 ft (230 m) on an edge, and the faces of triangles that are approximately equilateral.

The base of a pyramid can be any polygon of three or more edges, and pyramids are named according to the number of edges in the base. When the base is a triangle, the pyramid is a triangular pyramid. It is also known as a tetrahedron since, including the base, it has four faces. When these faces are equilateral triangles, it is a square pyramid, having a square as its base.

The pyramids most commonly encountered are "regular" pyramids. These have a regular polygon for a base and isosceles triangles for lateral faces. Not all pyramids are regular, however.

The height of a pyramid can be measured in two ways, from the vertex along a line perpendicular to the base and from the vertex along a line perpendicular to one of the edges of the base. This latter measure is called the slant height. Unless the lateral faces are congruent triangles, however, the slant height can vary from face to face and will have little meaning for the pyramid as a whole. Unless the word slant is included, the term height (or altitude) refers to the height.

If in addition to being congruent, the lateral faces are isosceles, the pyramid will be regular. In a regular pyramid, right triangles are to be found in abundance. Suppose we have a regular pyramid whose altitude is VC and slant height VD. Here the triangles VCD, VDE, VCE, and CDE are all right triangles. If in any of these triangle one knows two of the sides, one can use the Pythagorean theorem to figure out the third. This, in turn, can be used in other triangles to figure out still other unknown sides. For example, if a regular square pyramid has a slant height of two units and a base of two units on an edge, the lateral edges have to be √5 units and the altitude √3 units.

There are formulas for computing the lateral area and the total area of certain special pyramids, but in most instances it is easier to compute the areas of the various faces and add them up.

Volume is another matter. Figuring volume without a formula can be very difficult. Fortunately there is a rather remarkable formula dating back at least 2,300 years.

In Proposition 7 of Book XII of his Elements, Euclid showed that "Any prism which has a triangular base is divided into three pyramids equal to one another which have triangular bases." This means that each of the three pyramids into which the prism has been divided has one third the prism's volume. Since the volume of the prism is the area, B, of its base times its altitude, h: the volume of the pyramid is one third that, or Bh/3.

Pyramids whose bases are polygons of more than three sides can be divided into triangular pyramids and Euclid's formula applied to each. Then if B is the sum of the areas of the triangles into which the polygon has been divided, the total volume of the pyramid will again be Bh/3.

If one slices the top off a pyramid, one truncates it. If the slice is parallel to the base, the truncated pyramid is called a frustum. The volume of a frustum is given by the curious formula (B + B' + √BB )h/3, where B and B' are the areas of the upper and lower bases, and h is the perpendicular distance between them.

Resources

books

Eves, Howard. A Survey of Geometry. Boston: Allyn and Bacon, 1963.

J. Paul Moulton

KEY TERMS

Altitude

—The distance from the vertex, perpendicular to the base.

Pyramid

—A solid with a polygonal base and triangular lateral faces.

Slant height

—The distance form the vertex, perpendicular to the edge of the base.

pyr·a·mid / ˈpirəˌmid/ • n. 1. a monumental structure with a square or triangular base and sloping sides that meet in a point at the top, esp. one built of stone as a royal tomb in ancient Egypt.2. a thing, shape, or graph with such a form: the pyramid of the Matterhorn. ∎  Geom. a polyhedron of which one face is a polygon of any number of sides, and the other faces are triangles with a common vertex: a three-sided pyramid. ∎  a pile of things with such a form: a pyramid of logs. ∎  Anat. a structure of more or less pyramidal form, esp. in the brain or the renal medulla. ∎  an organization or system that is structured with fewer people or things at each level as one approaches the top: the lowest strata of the social pyramid. ∎  a system of financial growth achieved by a small initial investment, with subsequent investments being funded by using unrealized profits as collateral.• v. [tr.] heap or stack in the shape of a pyramid: debt was pyramided on top of unrealistic debt in an orgy of speculation. ∎  achieve a substantial return on (money or property) after making a small initial investment.DERIVATIVES: py·ram·i·dal / piˈramidl/ adj.py·ram·i·dal·ly adv.pyr·a·mid·i·cal / ˌpirəˈmidikəl/ adj.pyr·a·mid·i·cal·ly adv.

pyramid

pyramid. Monumental structure with a square base and steep battered triangular sides terminating in an apex. The form was used in Ancient Egypt for funerary structures: celebrated examples are the pyramids at Giza, near Cairo, Egypt (c.2551–c.2472 bc). Other types of pyramid include the stepped form found in both Ancient Egyptian and Meso-American Pre-Columbian architecture, but in the latter region the buildings were temple-platforms rather than tombs. The best-known stepped pyramids are the Ancient Egyptian pyramid at Saqqara, built by Imhotep for King Zoser (c.2630–c.2611 bc), and the temple-pyramids of the Meso-American Aztec and Maya cultures (c. C6–C16). Pyramids often featured in Neo-Classical architecture (Roman pyramids including that of Cestius (c.12 bc) ), and appealed to designers for their stereo-metrical purity of form, thus responding to Laugier's admiration for an architecture that was primitive and pure. Pyramidal compositions were common in funerary monuments from the time of Bernini, and countless C18 examples exist.

Bibliography

Carrott (1978);
J. Curl (2005);
Cruickshank (ed.) (1996);
I. Edwards (1985);
Fakhry (1969);
Gunnis (1968);
Hodges (1989);
Lepré (1990);
Lloyd & and Müller (1986);
W. S. Smith (1998);
Stadelmann (1985);
Jane Turner (1996);

pyramid a monumental structure with a square or triangular base and sloping sides that meet in a point at the top, especially one built of stone as a royal tomb in ancient Egypt.

Pyramids were built as tombs for Egyptian pharaohs from the 3rd dynasty (c.2649 bc) until c.1640 bc. The early step pyramid, with several levels and a flat top, developed into the true pyramid, such as the three largest at Giza near Cairo (the Pyramids, including the Great Pyramid of Cheops) which were one of the Seven Wonders of the World. Monuments of similar shape are associated with the Aztec and Maya civilizations of c.1200 bc–ad 750, and, like those in Egypt, were part of large ritual complexes.