area of a polygonal region

We consider only Euclidean geometryMathworldPlanetmath.

A triangular region is a triangleMathworldPlanetmath together with its interior. The sides of the triangle are called the edges of the region and the vertices of the triangle are called the vertices of the region.

A polygonal region is a plane figure that can be written as a union of a finite number of triangular regions, subject to a constraint. The constraint is that the triangular regions are nonoverlapping. That means that if two triangular regions intersect, then their intersectionMathworldPlanetmath is either a vertex or edge of each of the triangular regions.

A given polygonal region can of course be written as such a union in infinitely many ways.

We wish to assign a number to each polygonal region that corresponds to our intuitive idea of ”area”. To do that we first write down a set of postulatesMathworldPlanetmath that ”area” should satisfy and then show that the postulates can be satisfied. We let denote the set of polygonal regions.

  1. 1.

    There is a function α:.

  2. 2.


  3. 3.

    If two triangular regions are congruentPlanetmathPlanetmath, then they have the same area.

  4. 4.

    If two polygonal regions R1 and R2 are nonoverlapping then α(R1R2)=α(R1)+α(R2).

  5. 5.

    If a square has edges of length 1, then its area is 1.

The fifth postulate is needed because if α satisfies the first 4 postulates, then so does 2α. So, this postulate serves to make α a unique function.

Theorem 1.

If a square has sides of length s then its area is s2.

Proof. We first show that if a square has sides of length 1q then its area is 1q2. A unit square can be decomposed into q2 squares, each having side of length 1q. Each of the smaller squares has the same area, A, by postulates 3 and 4. Hence,


so that A=1q2. Next, if a square has sides of length pq then its area is p2q2. This is because the square can be written as p2 squares each having a side of length 1q. If A is the area then


Finally, let T be a square of side s and Tp/q denote a square of side pq having one angle in common with T. In the following sequencePlanetmathPlanetmath of statements, each statement is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to the next one:

  1. 1.


  2. 2.


  3. 3.


  4. 4.


  5. 5.


Hence, statements 1 and 5 are equivalent so that


so that s2=α(T). This proves the theoremMathworldPlanetmath.

One can then proceed on the basis of the postulates to show the following sequence of results.

Theorem 2.

The area of a rectangular region is the productPlanetmathPlanetmathPlanetmath of its base and its altitudeMathworldPlanetmath.

Rather than speak of the area of a triangular region or the area of a rectangleMathworldPlanetmathPlanetmath region, we shall refer to the area of a triangle, or area of a rectangle, or other polygonal region, as is usually done.

Theorem 3.

The area of a right triangle is 1/2 product of the lengths of its legs.

Theorem 4.

The area of a triangle is 1/2 the product of any base and the corresponding altitude.

Theorem 5.

The area of a parallelogramMathworldPlanetmath is the product of a base and the corresponding altitude.

Theorem 6.

The area of a trapezoidMathworldPlanetmath is 1/2 the product of the altitude and the sum of the bases.

Theorem 7.

If two triangles are similarMathworldPlanetmath then the ratio of their areas is the square of the ratio of any two corresponding sides.

What remains to do is to show that there is a function α that satisfies postulates 1 to 5. It might seem reasonable to just define the area of a rectangle, but rectangles are not useful for dissecting polygonal regions. The easier way to use triangles. So one defines the area of a triangle to be 1/2 the product of the base and its corresponding altitude. For this to be well-defined one has to show first that the product of a base and altitude (for a given triangle) does not depend on which base is chosen. Next, one defines the area of a polygonal region as the sum of the areas of the triangular regions in some decomposition. Of course one has to show that this sum does not depend on the particular decomposition that is used. Given this definition for α, it is then a simple matter to show that postulates 1 to 5 are satisfied.

The function α can be defined on a larger domain, which includes circular sectors and disks. But to do this rigorously ones needs the concept of a limit. In the hyperbolic geometry case, one can show the following theorem.

Theorem 8.

If α is a function that satisfies postulates 1 to 4 then there is a k>0 such that


for every polygonal region, where δ(R) is the defect of R.


  • 1 Edwin E. Moise, Elementary GeometryMathworldPlanetmath from an Advanced Standpoint, Third edition, Addison-Wesley, 1990
Title area of a polygonal region
Canonical name AreaOfAPolygonalRegion
Date of creation 2013-03-22 17:00:29
Last modified on 2013-03-22 17:00:29
Owner Mathprof (13753)
Last modified by Mathprof (13753)
Numerical id 12
Author Mathprof (13753)
Entry type Definition
Classification msc 51M25
Related topic Area2
Related topic InterceptTheorem
Related topic BaseAndHeightOfTriangle
Defines triangular region
Defines polygonal region
Defines area of a triangle