Area

The surface area is a measure of the size of an area. Lower surface is understood to be two-dimensional structures, that is those in which one can move in two independent directions. This includes the usual figures of plane geometry such as rectangles, polygons, circles, but also boundary surfaces of three-dimensional objects such as cubes, spheres, cylinders, etc. these areas are sufficient for many applications, complex surfaces can often be composed from these or approximated by this.

The area plays an important role in mathematics, the definition of many physical parameters, but also in everyday life. For example, pressure is defined as force per unit area or the magnetic moment of a conductor loop as current times umflossene surface. Land and apartment sizes are comparable by giving their base. Of material, for example of seed for a field or color for painting a surface, can be estimated by means of the surface area.

The area is normalized in the sense that the unit square that is the square with side length of 1, the area 1; expressed in units of measurement has a square with side length 1 m the surface area of ​​1 m2. In order to make comparable surfaces by their surface area, you should request that congruent faces have the same surface area and that the area of ​​composite surfaces is the sum of the contents of the patches.

The measurement of surface areas is not done directly in the rule. Instead, certain lengths are measured, from which the surface area is calculated. For the measurement of the surface area of a rectangle or of a spherical surface is usually measures the side lengths of the rectangle or the radius of the sphere, and has the desired surface area by means of geometrical formulas, as listed below.

Surface contents of some geometric figures

The table below shows some familiar characters from the planar geometry are listed together with formulas to calculate their surface content.

For the determination of the area of ​​a polygon can triangulate this, that is, by drawing diagonals into triangles disassemble, and then determine the areas of the triangles and finally add. Are the coordinates of the vertices of the polygon in a Cartesian coordinate system is known, the area can be calculated using the Gaussian trapezoid formula:

Here, and. The sum is positive if the vertices are traversed according to the rotation of the coordinate system. You may be to vote in the negative results of the amount. Especially for polygonal surfaces with lattice points as vertices can apply Pick's theorem. Other surfaces can be usually easily approximated by polygons, so that one can easily arrive at an approximate value.

Calculation of some surfaces

Here are some typical formulas for the calculation of surfaces are examples compiled:

A typical procedure for the determination of such surfaces is the so-called " rolling " or " unwinding " in the plane, and they are trying to map the surface in such a way in the plane such that the surface area is preserved, and then determines the surface area of the resulting flat figure. But that does not succeed at all surfaces, as the example of the sphere shows. For the determination of such surfaces methods of analysis are used in the example of the sphere can be employed as surfaces of revolution. Often the first Guldinsche usually leads to a rapid success, for example in the torus.

Integral calculus

The integral calculus was, among other things for the determination of surface areas under curves, that is, under function graphs developed. The idea is to approximate the area between the curve and axis by a series of narrow rectangles and then let the width of these rectangles go in a border case against 0. The convergence of this crossing point depends on the curve used. Considering a restricted area, such as the curve over a bounded interval as in the first drawing so show sets of analysis, that the continuity of the curve is already sufficient to ensure the convergence of the limit process. Here, the phenomenon occurs that space below the axis are negative, which may be undesirable in the determination of surface areas. If you want to avoid this, one must go on to the modulus of the function.

If you want to allow the interval limits and so we first determined the areas for finite limits and as just described and then lets in a further boundary process, or both aim. It may happen that this limit process does not converge, for example in case of oscillating functions such as the sine function. If we restrict ourselves to functions which have their function graph in the upper half-plane, these oscillation effects can not occur anymore, but it is not rare that the area between the curve and axis becomes infinite. As the total has an infinite extent, is a plausible and even ultimately expected result. If the curve, however, sufficiently rapidly approaches the axis for 0 far from remote locations, so the phenomenon can occur that also an infinitely extended surface plays a finite area. A well known and important for the probability theory example is the area between the Gaussian bell curve

And the axis. Although the area up enough, the surface area is equal to 1

In the attempt to further areas, such as under discontinuous curves to calculate, one finally comes to the question of which sets in the plane because to get even a reasonable area. This question is difficult, as is explained in the article to the measurement problem. It turns out that the intuitive area as used herein, can not be meaningfully extended to all subsets of the plane.

Differential Geometry

In differential geometry, the surface area of a flat or curved surface with the coordinates is calculated as a surface integral:

Here, the surface element of the interval width corresponds to the one-dimensional integral calculus. There is the area of ​​the plane defined by the tangents to the coordinate lines parallelogram with sides and. The surface element depends on the coordinate system, and the Gaussian curvature of the surface.

In Cartesian coordinates is the surface element. Is regarded as coordinate parameters on the spherical surface having the radius and the length and the width. For the surface of a sphere () is obtained so that the surface area:

For the calculation of the surface element, it is not absolutely necessary to know the position of a three-dimensional surface in space. The surface element can be derived solely from such dimensions that can be measured within the surface, and thus belongs to the inner geometry of the surface. This is also the reason that the area of ​​a ( developable ) surface does not change during unwinding and can thus be determined by unwinding in a plane.

Areas in physics

Surfaces naturally occur in physics as a quantity to be measured. Surfaces are usually measured indirectly by using the above formulas. Typical dimensions, where surfaces occur are:

• Pressure = force per unit area
• Intensity = energy per time and area
• Magnetic moment of a conductor loop = current times umflossene surface
• = Surface tension to increase the area worked per additional resulting surface
• Surface charge density = charge per unit area
• Current density = current per -carrying surface

Often, the area is also assigned a direction perpendicular to the surface, making the surface into a vector and gives it an orientation because of the two possible choices of the vertical direction. The length of the vector is a measure of the surface area. In a by vectors and limited parallelogram this is the vector product

If it is surfaces, one typically uses the normal vector field, in order to assign to the surface at any point in one direction can locally. This leads to flow - sizes is defined as the scalar product of the considered vector field and area ( as a vector). Thus, the current from the current density is calculated according to

Where in the integral the scalar product

Is formed. For the evaluation of such integrals are formulas for the calculation of surfaces helpful.

Occur in physics in addition also on area sizes that are actually determined experimentally, such as scattering cross sections. Here you go from the idea of a particle hits a solid target, the so-called target, and make the particles of the particle with a certain probability to the particles of the target. The macroscopically measured scattering behavior can then draw conclusions about the cross-sectional areas, which countered the target particles to stream species. The determined size has the dimension of an area. Since the scattering behavior not only of geometric parameters, but also by other interactions of the scatter depends partner with each other, the measured area is not always equate directly with the geometrical cross section of the scattering partners. One speaks of a general cross-section which also has the dimension of a surface.

Area calculation in surveying

Surface areas of land, land parts, countries or other areas can not be determined with the formulas for simple geometric figures in the rule. Such surface areas can be graphically semi graphically calculated from field measurements or coordinates.

In the graphical method, a mapping of the surface must be present. Areas whose boundaries are formed by a polygon can their basic lines and heights are measured, are decomposed into triangles or trapezoids. From these measurements it is the areas of the partial surfaces, and finally the area of ​​the total area can be calculated. The semi graphical area calculation is applied when the surface can be decomposed into small triangles whose short base side was measured in the field exactly. Since the relative error of the area is mainly determined by the relative error of the short base side, by measuring the base side in the field instead of the map, the accuracy of the surface area can be increased compared to the purely graphical method.

Irregular surfaces can be measured with a square glass plate. This carries on the underside of a grid of squares whose side length is known (for example 1 millimeter ). The tablet is placed on the mapped surface, and the surface area determined by counting the squares within the surface.

When extended surfaces a Planimeterharfe can be used. This consists of a sheet of parallel lines, whose distance is known uniform. The Planimeterharfe is placed on the surface such that the lines are approximately perpendicular to the longitudinal direction of the surface. Thus, the surface is divided into trapezoids whose center lines are added with a pair of dividers. The sum of the lengths of the center lines and the line distance of the surface area can be calculated.

Especially in areas with curvilinear boundary itself is the planimeter, a mechanical integrating instrument for investigation of the area. With the driving pin of the planimeter, the limitation must be worn. When driving around the area turns a role and a mechanical or electronic counter rotation of the pulley and the size of the area can be read. The accuracy depends on how accurate the editor with the tracer leaves the surface edge. The result is more accurate, the smaller the extent in proportion to the surface area.

The area calculation from field measurements can be applied when the surface can be decomposed into triangles and trapezoids and the data required for calculating surface distances are measured in the field. If the corners of the plot were aufgewinkelt in Orthogonalverfahren on the survey line, the area can also be calculated with the Gaussian trapezoidal formula.

Today surface areas are often calculated from coordinates. This can be, for example, the coordinates of boundary points in the real estate cadastre or vertices of a surface in a geographic information system. Often, the key points are occasionally connected by straight lines, also by circular arcs. Therefore, the surface area can be calculated using the Gaussian trapezoidal formula. For arcs, the circle segments between side of the polygon and arc to be considered. Is in a geographic information system of the contents of a more irregular surface to determine the surface can be approximated by a polygon with short side lengths.