Area axiom: Congruent polygons have the same area. The area of the sum of two polygons is the sum of their areas. The area of a rectangle is its base times its height.

Definitions: A polygon is a finite sequence of distinct points A₁, A₂, . . . , A_n and line segments s₁, s₂, . . . , s_n , so that s₁ = A₁A₂, s₂ = A₂A₃, . . . , s_n-1 = A_n-1A_n, s_n = A_nA₁, and the only intersection points of these segments are their common endpoints. The points A₁, A₂, . . . , A_n are called the vertices (singular: vertex) of the polygon, and the segments s₁, s₂, . . . , s_n are called the sides (or edges) of the polygon.

Two polygons are congruent if there is a bijection (one-to-one correspondence) between the vertices so that corresponding sides have equal lengths and corresponding angles have equal angle measures.

A triangle is a 3-sided polygon, and a quadrilateral is a 4-sided polygon. A rectangle is a quadrilateral so that the angles between adjacent sides are 90 degrees.

P + Q = R means roughly that R can be cut into two pieces P and Q. A careful definition involves stating a precise relation among the interiors of P, Q, and R: If P is a polygon, let P* denote the union of the sides of P and the interior of P. Using this notation, P + Q = R is defined to mean that (1) The union of P* and Q* is R*, and (2) The interiors of P and Q are disjoint.

Area is a function from the set of polygons to the set of positive real numbers. If P is a polygon, A(P) denotes the area of P.

Area axiom

1. Congruence: For all polygons P and Q, if P is congruent to Q, then A(P) = A(Q).

2. Additivity: For all polygons P, Q, R, if P + Q = R, then A(P) + A(Q) = A(R).

3. Rectangle: The area of a rectangle is the product of the lengths of two adjacent sides.

Comment on areas of general regions

The area of a polygon is really the area of the interior of the polygon. (In Geometer's Sketchpad, for example, to measure the area of a polygon first you have to construct its interior and then measure the area of the interior.) The interior of a polygon is often called a "polygonal region."

To understand the areas of more general regions (subsets) of the plane, one more property of area is needed.

4. Containment: If the set X is contained in the set Y, then A(X) ≤ A(Y).

It follows from this property that the area of a set X - if it is defined - must be the limit of the areas of all the polygons with interiors contained in X. (More precisely, A(X) is the least upper bound of all the real numbers A(P) such that P is a polygon with the interior of P contained in X. The area A(X) is defined if this least upper bound is equal to the greatest lower bound of all the real numbers A(P) such that X is contained in the interior of P.)

This principle was used by the ancient Greeks to compute the area of a circle and other regions. Archimedes used the analogous principle to compute the volume of a sphere, and of many other interesting solids. (See the fascinating book Archimedes: What Did He Do Besides Cry Eureka? by Sherman Stein, Mathematical Association of America, 1999.)

A simpler use of limits is the definition of the length of a curve. The length of a curve C is the limit of the lengths of polygonal arcs P inscribed in C, i.e. so that all the vertexes of P lie on C. This limit is taken as the maximum length of the segments of P goes to zero. Using this definition it's easy to see that the circumference of a circle is proportional to its diameter. The Greeks named the ratio of the circumference to the diameter pi.

You may recall from calculus that the length of a curve can be computed using derivatives and integrals. (In physical terms, the distance travelled by a moving object is the integral of its speed.) The uses of calculus to do geometry are explored in the course Differential Geometry (MATH 4250/6250).

You certainly should remember that integrals are used in calculus to compute areas. A full investigation of the theory of area and volume is contained in the course The Lebesgue Integral and Its Applications (MATH 4110/6110).

Comment on the area axiom

Using limits and the properties of real numbers the area axiom can be simplified. The formula (3) for the area of a rectangle can be replaced by the simple property that the area of a unit square is 1. (A unit square is a square with side 1.)