Area in the plane

We consider areas of plane sets bounded by closed curves. In the more general cases, the concept of area becomes theoretically very difficult.

The area of a planar set is defined by reducing to the areas of simpler sets. The area cannot be "calculated", unless we first have a definition of "area" (although this is common practice in school mathematics).

Starting point

The area of a rectangle
The area of a rectangle is base \(\times\) height: \[A=ab.\]

A rectangle
Definition: Area of a Parallelogram

The area of a parallelogram is base \(\times\) height: \[ A=ah. \]


Parallelogram
Definition: Area of a triangle

The area of a triangle is (by definition) \[ A=\frac{1}{2}ah. \]


Triangle

Polygon

A (simple) polygon is a plane set bounded by a closed curve that consists of a finite number of line segments without self-intersections.

Polygon
Definition: Area of a polygon

The area of a polygon is defined by dividing it into a finite number of triangles (called a triangulation of the polygon) and adding the areas of these triangles.


Triangulation
Theorem.

The sum of the areas of  triangles in a triangulation of a polygon is the same for all triangulations.


General case

For a plane set \(\color{red} D\) bounded by a closed curve we can construct inner polygons \(\color{blue}P_i\) and outer polygons \(P_o\): \(\color{blue}P_i\color{black} \subset \color{red}D\color{black}\subset P_o\).

A bounded set \(D\) has an area if for every \(\varepsilon >0\) there is an inner polygon \(P_i\) and an outer polygon \(P_o\), whose areas differ by less than \(\varepsilon\): \[ A(P_o)-A(P_i)<\varepsilon. \] This implies that between all areas \(A(P_i)\) and \(A(P_o)\) there is a unique real number \(A(D)\), which is (by definition) the area of \(D\).

Inner and outer polygons

A surprise: The condition that \(D\) is bounded by a closed curve (without self-intersections) does not guarantee that it has an area! Reason: The boundary curve can be so "wiggly", that it has positive "area". The first such example was constucted by [W.F. Osgood, 1903]:

Wikipedia: Osgood curve

Example

Derive the formula \(A=\pi R^2\) for a circle with radius \(R\) by choosing regular inscrided and circumscribed \(n\)-gons as inner and outer polygons, and let \(n\to\infty\).

The solution is a voluntary exercise, where you need the limit \[\lim_{x\to 0}\frac{\sin x}{x} = 1.\] Hint: Show that the inscribed and circumscribed areas are \[ \pi R^2\frac{\sin (2\pi/n)}{2\pi/n} \ \text{ and }\ \pi R^2\frac{\tan \pi/n}{\pi/n}.\]

Last modified: Thursday, 28 January 2021, 1:59 PM