﻿ Pyramid (geometry)

# Pyramid (geometry)

The pyramid is a body of the geometry. This polyhedron comprises a plurality of seamless contiguous flat surfaces, one of which is a polygon and all other triangles. The triangles ( faces) form the lateral surface.

• 4.1 lateral surface calculation ( square pyramid )
• 7.1 Equations
• 7.2 Elementary Geometric reasoning
• 7.3 grounds with the help of integral calculus

## Construction

From an excellent point, the pyramid, is from a bundle of rays that intersect the rays of a level in the corners of the base of the pyramid. Four rays of a certain inclination in the room is obtained, for example, a square base, thus forming the square pyramid. It is the construction with an arbitrary area of ​​a polygon (polygon ) of the plane start and a point (this is the top of the pyramid are ) choose outside this plane. By connecting each vertex of the base to the tip, created the aforementioned bundle of rays. The points of each base edge are connected through the triangular area with the apex of the pyramid. Thus, the pyramid meets the definition of a cone.

## Properties

If the base of a pyramid n vertices, then the number of the ( triangular ) faces also equal to n, together with the base, the pyramid then a total of n 1 surfaces. The number of corners is now but also n 1, namely n corners of the base and the tip. Finally, bring the base n edges and the side lines of the pencil of rays that connect the corners of the base to the apex of the pyramid, just as many, making a total of 2n edges. Thus, the Euler Polyedersatz on the numbers of corners ( e), met surfaces ( f) and edges (k):

For the calculation of the pyramid volume ( see below), the concept of height is important. It refers to the ( shortest ) distance between the tip of the plane in which the base is located.

## Deferrals and special cases

### Regular (regular ) pyramid

From a periodic or regular pyramid is when the base is a regular polygon and the center of this polygon is also the base of the pyramid height. Each regular pyramid is therefore just. Among the regular pyramids addition to those mentioned below regular tetrahedra and the square pyramids. They have a square as a base, wherein the link between the square center, and the pyramid tip is perpendicular to the base surface.

The focus of a pyramid is generally on the link between the centroid of the base and top of the pyramid. It divides this range in the ratio 1:3, and therefore the distance from the base.

### Especially pyramid

A pyramid with a regular polygon (polygon ) as the base is called even if the base has a center M, all lateral edges ( ie all edges emanating from the top ) are of equal length, the link between M and the peak S perpendicular to the base of the pyramid and thus the foot of the perpendicular from the center S with a center point of the base is identical, ie, lies in the interior of the base.

If the base is not regular, but at least a point- symmetric polygon, we can speak of a right pyramid even if the center of symmetry of this polygon coincides with the Höhenfußpunkt the pyramid. The side edges, however, can be of different lengths.

If the base of a pyramid is neither a regular nor a point- symmetric polygon, then the term has just no meaningful significance more: Is the base, for example an arbitrary triangle, the apex of the pyramid should be vertically above its circumcenter, so that all page edges are of equal length. If this triangle is more obtuse, then the nadir point of the tip is even outside the base - which contradicts the ( ideological ) importance of reading.

### Leaning pyramid

A pyramid with a regular polygon (polygon ) as the base is called crooked if not all page edges are of equal length, the foot of the perpendicular from the point S is not the center M of the base and therefore the link from the center of the base M and the peak S is not perpendicular to the base of the pyramid. In a lopsided pyramid of the foot of the perpendicular from the point S can be located, therefore, both inside and outside of the pyramid base.

The volume can be calculated like a regular pyramid, so: or.

### Johnson - body J1

Is a square pyramid straight and its four side surfaces equilateral triangles, the square pyramid is thus also the easiest of the Johnson - body, abbreviated as J1.

### Tetrahedron

A pyramid as the base has an equilateral triangle and its three side surfaces are also equilateral triangles congruent to the base is called (regular ) tetrahedron ( tetrahedron ). If you would upset her, she would still look the same as before.

## Surface calculation ( square pyramid )

The surface of a quadrangular pyramid is composed of the square base (G) and the shell ( M)

Is the page length (a ) where, leads to the following formula:

### Lateral surface calculation ( square pyramid )

In a regular pyramid with a square base, the lateral surface of the four areas is congruent, isosceles or possibly together also equilateral triangles. See also.

If the side length ( a) and if the pyramid height (h), then the following formulas or equations solution found:

The area of ​​these triangles is, all four faces, saying, or after transformation:

Here, the height of the side congruent triangles.

From the Pythagorean theorem gives:

It follows: and thus for the total shell surface: or by forming:

## Length calculation of the steep edges ( square pyramid )

In addition to the four base edges ( a), which are identical with the side length, has the square pyramid four equally long steep edge also called burrs (AS ), ( BS ), ( CS ) and ( DS), which extend from the corners of the base and upward sloping in the pyramid tip ( S) meet.

If the side length ( a) and if the pyramid height (h), then the following formulas or equations solution found:

First, the length of the base plane diagonal ( d) must be calculated. This follows from the Pythagorean theorem: it follows:

For the further calculation you need the half of ( d ), ie: is then the square and it is by forming

For the calculation of AS using again the Pythagorean Theorem: and it follows then for the ridge

## Calculation of the total edge length ( square pyramid )

The total edge length of the square pyramid (K ) consists of the four side lengths together (a) and the four equally long ridges (AS ), ( BS ), ( CS) and (DS). In turn, the side length (a) and the pyramid height given ( h), the result for the total edge length following equation solution:

Or after transformation

## Volume calculation

### Formula

The volume V of a pyramid is calculated from the contents of the base ( G) and the height ( h) in accordance with

This formula applies to each pyramid. So it does not matter if the base of a triangle, square, pentagon, ... is. The formula is also valid if the Höhenfußpunkt does not coincide with the base or the base surface center has no center. In the special case of a square pyramid is obtained, where A is the side length of the square base, and h is the height.

The general formula corresponds to the way the volume formula for a circular cone. This is because that each pyramid meet the definition of a general cone. Conversely, a cone to be construed as a pyramid with a regular n-polygon as a base, which is degenerate to a circle by n → ∞.

### Elementary geometric reasoning

The above volume formula can be elementary geometric reasons in three steps:

• 2 For pyramids with a triangular base of the volume formula applies.
• 3 The volume formula is valid for any pyramid.
• 4 A cube can be divided into 3 equal pyramids with a square base, the tips terminate at a corner of the cube.

### Reasoning with the help of integral calculus

The volume of a pyramid with the base G and height h can be calculated when the pyramid of thin ( infinitesimal ) layers of thickness dy imagines constructed parallel to the base. A y-axis is now put through the top of the pyramid, so that the height h coincide with the y-axis. Denoting the surface of the layer at a distance y from the tip, so you can from the laws of central dilation derive a formula for:

The volume of a layer is then dV = A ( y) dy. Finally, the volume of the pyramid is the sum of the volumes of all individual layers. This sum is obtained by integration of y = 0 to y = h

## Volume square pyramids as an extreme value

The ball, cube or regular tetrahedron body whose volume is maximum, for a given surface area, ie any change in the outer shape would give a smaller volume. The same is true for the regular octahedron, which can be understood as two composite with the base sides square pyramids from each 4 equilateral triangles. A square pyramid with maximum volume is comparatively sharp. Refers to the base naturally in the total surface area of a pyramid with a, we obtain for such a pyramid with base side length a and a height of a volume, the triangles of the lateral surface having a height of.

The figure on the right represents a square pyramid whose base fourth corner not be detected in perspective distortion by receiving position and lens choice.

## Measurement of a pyramid structure

With a large pyramid, the edge lengths of the base can be directly measured accurately, but not the height that is not directly accessible. The following are the basic difficulties are to be explained, not related so much with the methodology of the measurement process itself. A simple geometric method to assess the height of larger objects is to look from a distance and the determination of visual angle. In simplified form by the following graphic shown:

At a distance s from the lower edge of the pyramid top of the pyramid is targeted α under the measured angle. The distance of the observation point of the pyramid in a horizontal line is thus half the base side a / 2 s The height h is obtained from the formula in the graph. In order to determine the amount would be a minor problem. However, the following difficulties:

• The top of the pyramid is not necessarily exactly over the center of the base
• The length of the base edge of the pyramid is not clean determinable ( broken stones, erosion)
• The tip is no longer available ( removed )
• The angle of inclination of the pyramid is elusive ( removal, erosion)

This largely corresponds to the realities of the famous Great Pyramids. The amount of deviation of the observation point at which α is measured must be taken into account exactly. The angle measurement itself can be made very accurate in general. Furthermore, it must be defined, from which the floor level of the height of the pyramid is to be valid, that is where they should actually start. Assuming that the base of the pyramid could be a length of no more accurate than 30 cm, and thus the distance a / 2 s do not specify the measuring point more accurately than at 15 cm. This would amount to the amount of from about 10 cm to be inaccurate at a viewing angle α of 35 ° adopted. It is now missing still determining the angle of inclination β of the side surface. A hypothetical large pyramid base length of 200 m and a height of 140 m would have in an inaccuracy in the altitude of 10 cm inaccuracy of the inclination angle data of about 1 minute of arc (54 ° 27'44 " at h = 140.0 m compared to 54 ° 26 '34 ' with h = 139.9 m). This now applies to pyramids whose peak is still present. The reality is different. The amount of provision is not there is the original height again, but the amount of ablated pyramid.

Thus, the tip must be extrapolated. The following figure shows schematically the problem. Both the sides and the top are significantly removed by demolition and weathering:

The height h of β would be accessible according to the formula from the direct determination of the angle of inclination. As can be seen, the determination is subject to large errors. An exception is the Pyramid of Chephren, because this still has the original deck stones in the upper part. The angle β is thereby determined more accurately than the other pyramid. Which explains the good match with regard to the angle of inclination of the various authors.

It is thus clear that in real pyramids neither the height to the nearest centimeter nor the inclination angle can be specified precisely to the second of arc.

## Related terms

Other geometric solids, which are closely related to the concept of the pyramid, are the truncated pyramid ( a parallel to the base " truncated " pyramid) and the bipyramid, which is composed of two pyramids. With the pyramid in the architecture, the article focuses pyramid (Building).

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