Home > Area (geometry)

1 How to define area

Although area seems to be one of the basic notions in geometry it is not at all easy to define, even in the Euclidean plane. Most books avoid it, wrongly referring to self-evidence. To make area meaningful one has to define it, at the very least, on polygons in the Euclidean plane, and it can be done using the following definition:

The area of a polygon in the Euclidean plane is a positive number such that:

1. The area of the unit square is equal to one.
2. Equal polygons have equal area.
3. If a polygon is a union of two polygons which do not have common interior points, then its area is the sum of the areas of these polygons.

But before using this definition one has to prove that such an area indeed exists.

In other words, one can also give a formula for the area of an arbitrary triangle, and then define the area of an arbitrary polygon using the idea that the area of a union of polygons (without common interior points) is the sum of the areas of its pieces. But then it is not easy to show that such area does not depend on the way you break the polygon into pieces.

Nowadays, the most standard (correct) way to introduce area is through the more advanced notion of Lebesgue measure, but one should note that in general, if one adopts the axiom of choice then it is possible to prove that there are some shapes whose Lebesgue measure cannot be meaningfully defined. Such 'shapes' (they cannot a fortiori be simply visualised) enter into Tarski's circle-squaring problem (and, moving to three dimensions, in the Banach-Tarski paradox). The sets involved do not arise in practical matters.

2 Some formulae

For a two dimensional object the area and surface area are the same:

• square or rectangle: (where l is the length and w is the width; in the case of a square, l = w.
• circle: (where r is the radius)
• ellipse: (where a and b are the semi-major and semi-minor axes)
• any regular polygon: (where P = the length of the perimeter, and a is the length of the apothem of the polygon [the distance from the center of the polygon to the center of one side])
• a parallelogram: (where the base B is any side, and the height h is the distance between the lines that the sides of length B lie on)
• a trapezoid: (B and b are the lengths of the parallel sides, and h is the distance between the lines on which the parallel sides lie)
• a triangle: (where B is any side, and h is the distance from the line on which B lies to the other vertex of the triangle). This formula can be used if the height h is known. If the lengths of the three sides are known then Heron's formula can be used: (where a, b, c are the sides of the triangle, and is half of its perimeter)
• the area between the graphIn mathematics, the graph of a function is the collection of all pairs x f ''x ) of the function. In particular graph means the graphical representation of this collection, in the form of a curve, together with axes, etc. Graphing is sometimes referred tos of two functions is equalSee also the disambiguation page title equality. In mathematics, two mathematical objects are considered equal if they are precisely the same in every way. This defines a binary predicate, equality denoted " "; x y iff x and y are equal. Equivalence in th to the integralThis article deals with the concept of an integral in mathematical calculus. For other meanings of "integral" see integration. In calculus, the integral of a function is a generalization of area, mass, volume, sum, and total. Unlike the process of differe of one functionIn mathematics, a function is a relation such that each element of a set (the domain is associated with a unique element of another (possibly the same) set (the codomain not to be confused with the range . The concept of a function is fundamental to virtu, f(x), minusIn mathematics, subtraction is one of the four basic arithmetic operations. It is usually denoted by an infix minus sign. The traditional names for the terms of the subtraction c − b a are difference a , minuend (c) and subtrahend (b). Basic subtrac the integral of the other function, g(x).

Some basic formulas for calculating surface areas of three dimensional objects are:

• cubeThree dimensions A cube (or hexahedron is a Platonic solid composed of six square faces, with three meeting at each vertex. The cube is a special kind of square prism, of rectangular parallelepiped and of triangular trapezohedron, and is dual to the octah: , where s is the length of any side
• rectangular box: , where l, w, and h are the length, width, and height of the box
• sphereFor other uses, see sphere (disambiguation). A sphere is, roughly speaking, a ball-shaped object. In non-mathematical usage a sphere is often considered to be solid (which mathematicians call ball . But in mathematics, a sphere is the boundary of a ball,: , where π is the ratio of circumference to diameter of a circle, 3.14159..., and r is the radius of the sphere
• ellipsoid: see the article
• cylinder: , where r is the radius of the circular base, and h is the height
• cone: , where r is the radius of the circular base, and h is the height.