Area Estimates by Monte Carlo  Simulation

Objective: To estimate the area of a circle or triangle using a probability experiment employing the Monte Carlo technique. We also indicate how to use our approach to estimate the area of a polygonal region. (For a general description of the Monte Carlo technique go to the Elementary Probability Demo Collection.)

Level: High school, precalculus, or probability courses for math majors.

Prerequisites: Basic probability concepts; knowledge of equally likely chance and the chance of winning a lottery.

Platform: A calculator or computer which has a function that generates random numbers or alternatively a phone book. Several MATLAB routines are used for illustrations of the technique.

Instructor's Notes:

We start with a typical problem that can be solved by geometry or estimated by Monte Carlo simulation experiments. We extend the solution of this basic problem to the estimation of the area of other geometric figures.

Find the probability p of selecting a point on or in the interior of a circle of radius 1 which is inscribed in a square with side 2. (Assume that the circle is centered at the origin.) 
To use Monte Carlo simulation we use a random number generator to obtain ordered pairs (x, y) where -1  1 and -1  1. If  then (x, y) is in the circle. When a large number of pairs (x ,y) are chosen, then the ratio
approximates the probability p. It follows that the area of the circle is approximately .

An interesting procedure for generating the random numbers is to use a telephone directory. The last two, three, or four digits of the telephone numbers in any column of any page can be used as a coordinate of a point in the square. (The first three digits of a telephone number represent a locality and are not random.) Using the last four digits a decimal between 1 and -1 is encoded as follows:

If the leading digit is even, then the decimal is taken to be positive; otherwise it is negative. The remaining three digits are used to form a decimal of the style .
Having generated an x-coordinate as described above, repeat the procedure on the phone number in the adjacent column of the page to obtain the corresponding y-coordinate. Naturally we could use calculators or computers to generate the pairs (x, y). However, the phone book generation enhances the hands-on aspect and seems to stimulate a real involvement with the overall process.

This experiment works very well using collaborative groups. Each group selects a page from the telephone directory and two adjacent columns. Students have enjoyed this approach and there is an opportunity for this to evolve into team projects. Such extensions may involve areas of polygonal figures, going to 3-dimensions (a sphere within a cube), or going further to n-dimensions (an n-ball in an n-dimensional hypercube).

The animation in Figure 1 illustrates the probability computation described above. This animation was created using a MATLAB routine  which can be downloaded  by clicking on montecirc.ZIP . (This routine lets the user select the number of random trials to use for an experiment.) Table 1 presents the approximations for a set of such experiments.

                                                                     Figure 1.

                                                           Table 1.

A triangle is a familiar polygonal region  whose area can be approximated by a Monte Carlo process. The animation in Figure 2 illustrates such an approximation. This animation was created using a MATLAB routine  which can be downloaded  by clicking on montetri.ZIP . (This routine lets the user select the vertices of the triangle either by entering three distinct ordered pairs or using a mouse generate the triangle. In addition the number of random trials to use for an experiment can be specified.) The approximation generated in the animation is 13 square units, which is actually quite close to the exact area, 13.5. The triangle shown in Figure 2 has vertices A(-1,3) B(4,1), and C(1,-2). Other experiments with this triangle need not produce as accurate an approximation even if we increase the number of trials.  Table 2 presents the approximations for such a set of such experiments. 

                                                         Figure 2.

                                                         Table 2.

From the data in Table 2 we infer that in order to have confidence in the approximation generated from a Monte Carlo simulation, a large number of trials are needed. We ran 100 experiments for the triangle in the animation in Figure 2 using k = 1000. All estimations were between 8.1 and 12.6 with an average of 10.636, yet the exact area is 13.5. A natural question is, how many experiments should be used and how many  trials should be used? If you get this question, your students have tuned in to this modeling problem and are ready for more ideas in probability theory. 

Another question arises when you compare the circle simulation with the triangle simulation. Since we know that (x, y) where -1  1 and
-1  1 is in the circle provided , how was it determined whether (x, y) where -5  5 and -5  5 is in the triangle ABC?
We don't have an equation for the triangle (in usual sense), so how do test the coordinates (x ,y)? For beginning classes this may be difficult to answer, but it is a good question for students to write about. It is possible to relate the situation to inequalities (click on inequalitydemo for foundational material) or linear algebra with systems of equations and parametric representations of lines. For advanced classes the notion of  the convex hull of a set a points can be used. In fact for polygons, the notion of the convex hull is very useful for determining whether or not (x, y) is within the polygon.

Other resources.

          A nice web-based Monte Carlo simulation for approximating the area of a circle
          is available at

          Use the key phrase geometric probability in the search feature to find a gizmo 
          related to this demo. On this site students are able generalize the area example
          by changing the size of the circle and a surrounding rectangle.

          For a more general setting for area estimation under a curve go 
          to  Monte Carlo Demo

Credits:  A portion of this demo was adapted from the following work.
Regina Brunner, "Numbers, Please! The Telephone Directory and Probability", The Mathematics Teacher, Vol. 90, No. 9, Dec. 1997, pp. 704 -705.
and we appreciate the cooperation of 
Regina Brunner
Assistant to the Provost for Research and Planning
Kutztown University
in its development. 

We also acknowledge contributions by Un Jung Sin , student at Temple University. The MATLAB routines were written by David R. Hill.

DRH 6/2/01   Last updated   5/22/2006

Since 6/14/01